Recent research findings on M87 (NGC 4486)

M87 (Messier 87), also known as NGC 4486, is a giant elliptical galaxy, located about 53.5 million light-years away. It is noteworthy for several reasons, including the presence of an unusually large supermassive black hole (SMBH) in its active galactic nucleus, with an estimated mass of about 6.4×10⁹ times the mass of the Sun (M_⊙), two plasma jets that emit strongly at radio frequencies and extend at least 5000 light-years from the SMBH (although only the jet pointed more towards us is readily detectable), and a population of about 15,000 globular clusters.

The total mass of M87 is difficult to estimate, because elliptical galaxies like M87, and unlike spiral galaxies, do not tend to follow the Tully-Fisher relation between intrinsic luminosity and total mass calculated from rotation curves – which therefore includes dark matter. Estimates of the total mass of M87, including dark matter, come in around 6×10¹² M_⊙ within a radius of 150,000 light-years from the center. This compares with about 7×10¹¹ M_⊙ for the Milky Way, but M87 could be more than 10 times as massive.

In other comparisons, the Milky Way has only about 160 globular clusters, and a central black hole (Sagittarius A*) with a mass of about 4.2×10⁶ M_⊙. So M87's central black hole is about 1500 times as massive as the Milky Way's. Pretty impressive difference.



M87 – click for 640×480 image

Besides the recent research listed below, I've written about earlier research on M87 in these articles: Galactic black holes may be more massive than thought, Stellar birth control by supermassive black holes, Black holes in the news.

You might also be interested in some articles from the past year on the general subject of active galaxies: Active galaxies and supermassive black hole jets, Where the action is in black hole jets, Quasars in the very early universe.

Feedback under the microscope: thermodynamic structure and AGN driven shocks in M87 (6/29/10) – arXiv paper

Feedback under the microscope II: heating, gas uplift, and mixing in the nearest cluster core (3/28/10) – arXiv paper

Activity of the SMBH in M87 has a significant effect not only on the host galaxy, but also on the Virgo cluster of galaxies in which M87 is near the center. Energetic outflows of matter from near the black hole force plumes of gas out of the galaxy into the hotter intergalactic medium. The mass transported in this way represents about as much gas as is contained within 12,000 light-years of M87's center. (However, that's only about 2.5% of M87's 500,000 light-year radius.) If it had not been expelled, the gas could have formed hundreds of millions of stars.

The first paper reports on studies using the Chandra X-ray Observatory to measure gas temperatures around M87's center. The findings include detection of 2 distinct shock wave fronts about 46 thousand light-years and 10 thousand light years from the center. This indicates that explosive events occurred about 150 million and 11 million years ago, respectively.

The second paper uses observations from Chandra, XMM-Newton, and optical spectra to distinguish different phases of the hot gas surrounding M87's SMBH.

Refs:
• Galactic 'Super-Volcano' in Action (8/20/10) – Science Daily (press release)
• Galactic Supervolcano Erupts From Black Hole (8/20/10) – Wired.com
• Galactic 'Supervolcano' Seen Erupting With X-Rays (9/6/10) – Space.com

A correlation between central supermassive black holes and the globular cluster systems of early-type galaxies (8/13/10) – arXiv paper

A study of 13 galaxies, including M87, has found a correlation between the size of a galaxy's SMBH and the number of the galaxy's globular clusters. The types of galaxies studied included nine giant ellipticals (like M87), a tight spiral, and 3 galaxies intermediate in type between spiral and elliptical. The smallness of the sample is due to the exclusion of open spiral galaxies and the further limitation to cases where good estimates of the number of globular clusters and mass of the central black hole existed.

The correlation, in which the number of globular clusters is proportional to the black hole mass, is actually stronger than correlations between black hole mass and other galaxy properties previously studied for correlation, such as stellar velocity dispersion (an indicator of total mass), and luminosity of the galaxy's central bulge or whole galaxy (for ellipticals).

In some cases the correlation of black hole mass with total luminosity was especially weak, but better with number of globular clusters. For instance, Fornax A (NGC 1316) is a giant lenticular galaxy with luminosity comparable to that of M87. Yet its central black hole has a mass of 1.5×10⁸ M_⊙, 2.3% that of M87's black hole. It has 1200 globular clusters, 8% of M87's count. Clearly this is not a linear relation. Rather, the study found that the best fit was a power law with M_• ≈ (1.7×10⁵)×N^1.08±0.04, where M_• is black hole mass in units of M_⊙ and N is number of globular clusters. This relation predicts a SMBH mass of 5.5×10⁹ M_⊙ for M87, which is very close, and 3.6×10⁸ M_⊙ for the SMBH mass of NGC 1316, which is high – but the SMBH mass of NGC 1316 is also unusually low in comparison with its luminosity and velocity dispersion.

By contrast, the relation predicts that the Milky Way with a SMBH mass of 4.2×10⁶ M_⊙ should have only about 20 globular clusters, while the actual number is about 160. However, the Milky Way is a loose spiral, not one of the types that was studied, which may account for the much worse correlation. The fit is much better if only globular clusters associated with the central bulge (about 30) are considered.

The obvious question is about why this relation between SMBH mass and number of globular clusters exists. Presumably it has much to do with the typical history of a large galaxy, which is expected to include frequent mergers with other galaxies. The existence of the relationship should provide clues to galactic history, and especially how this may be different for loose spirals like the Milky Way, in comparison with more compact galaxies.

Refs:
• A correlation between central supermassive black holes and the globular cluster systems of early type galaxies (8/11/10) – The Astrophysical Journal
• Supermassive black holes reveal a surprising clue (5/25/10) – Physicsworld.com

A Displaced Supermassive Black Hole in M87 (6/16/10) – arXiv paper

It has generally been assumed that a galaxy's central SMBH is very close to the actual center of mass of the galaxy, because that is (by definition) the gravitational equilibrium point. This central point should be essentially the same as the photometric center of the galaxy, since the galaxy's stars should be distributed symmetrically around the center. Consequently, astronomers have not carefully searched for cases where a SMBH is not very near the galactic center. This lack of extensive investigation is also a result of the fact that the SMBH is often hidden inside a dense cloud of dust, so its exact position is difficult to determine. M87's SMBH (more precisely, the accretion disk around the SMBH), however, is clearly visible, and the research reported in this paper finds it is actually located about 22 light-years from the apparent galactic center.

There are various possible reasons for this much displacement from the center, and not a lot of evidence to identify the most likely reason. Possible reasons include: (1) The SMBH is part of a binary system in which the other member is not detected. (2) The SMBH could have been gravitationally perturbed by a massive object such as a globular cluster. (3) There is a significant asymmetry of the jets. (4) The SMBH has relatively recently merged with another SMBH, subsequent to an earlier merger of another galaxy with M87.

The displacement of the SMBH is in the direction opposite the visible jet, so the last two possibilities are more likely than the others. However, possibility (3) depends on the jet structure having existed at least 100 million years and the density of matter at the center of M87 being low enough to provide insufficient restoring force. Possibility (4) is viable if the SMBH is still oscillating around the center following a galactic merger within the past billion years.

Refs:
• A Displaced Supermassive Black Hole in M87 (6/9/10) – The Astrophysical Journal Letters
• Black Hole Shoved Aside, Along With 'central' Dogma (5/25/10) – Science News
• Black Hole Found in Unexpected Place (5/25/10) – Wired.com
• Supermassive black holes may frequently roam galaxy centers (5/25/10) – Physog.com (press release)
• Bizarre Behavior of Two Giant Black Holes Surprises Scientists (5/25/10) – Space.com
• Galactic Black Holes Can Migrate or Quickly Awaken from Quiescence (5/26/10) – Scientific American




M87 jet

Radio Imaging of the Very-High-Energy γ-Ray Emission Region in the Central Engine of a Radio Galaxy (7/24/09) – Science

Energetic plasma jets, in which matter is accelerated close to the speed of light, combined with intense electromagnetic emissions, especially at radio frequencies, are prominent in about 10% of active galaxies, including M87. However, little has been well established about what processes are responsible for the emissions, or more generally how the jets are powered, accelerated, and focused into narrow beams. Because of the relative proximity of M87 and the fact that the jet we observe is angled from 15° to 25° to our line of sight, M87 is one of the best objects to study in order to learn more about how jets work.

Gamma rays, because of their very high energies (greater than 100 keV per photon), are not continuously produced in active galaxy jets, but are occasionally observed in short bursts lasting only a few days. One such event occurred in M87 in February 2008. At the same time, the intensity of radiation at all other wavelengths increased substantially. Such flares, at lower energies, are not unusual, since the energy output of most jets is somewhat variable in time. The flare persisted for much longer at energies below the gamma-ray band, indicating that the disturbance continued to propagate along the jet even after the gamma-ray flare subsided. However, although we don't know what the cause was, the coincidence in time of the gamma-ray emissions and the beginning of the extended flare makes it very likely that the events had the same source.

This is significant information, because our technology for detecting gamma-ray events has very poor angular resolution (~0.1°), since gamma rays can be detected on the ground only by secondary effects that a gamma ray produces in our upper atmosphere. More than 6 orders of magnitude finer resolution can be achieved at radio frequencies, using very long baseline interferometry. With that technology, it was possible to locate the origin of the disturbance that caused both gamma ray and lower energy flaring to a region within about 100 Schwarzschild radii (R_s) of the SMBH. Since R_s = 2G×M_•/c², R_s for the M87 SMBH is about 1.9×10¹⁰ km, or more than twice the radius of the solar system. So 100R_s is about 70 light-days – which is pretty small compared to the 53.5 million light-year distance to M87.

It's also significant that the gamma-ray event occurred so close to the SMBH, because the cause must be unlike whatever is responsible for the flaring described in the following research.

Refs:
• VLBA locates superenergetic bursts near giant black hole (7/2/09) – Physorg.com (press release)
• Mysterious Light Originates Near A Galaxy's Black Hole (7/2/09) – Space.com
• A Flare for Acceleration (7/24/09) – Science
• High Energy Galactic Particle Accelerator Located (9/14/09) – Science Daily (press release)

Hubble Space Telescope observations of an extraordinary flare in the M87 jet (4/22/09) – arXiv paper

Electromagnetic radiation from SMBH jets is fairly variable in both time and location along the jet. In the case of M87, high-resolution images at various wavelengths have shown the existence of many regions of enhanced emissions within the jet. One of the most prominent of these even has a name: HST-1, so-named because it was discovered by the Hubble Space Telescope. It occupies a stationary position on the jet, about a million Schwarzschild radii from the center, i. e. about 2000 light-years from the SMBH.

HST-1 has been observable for some time, but until February 2000 it was relatively dormant. After that it began to flare more brightly across the electromagnetic spectrum up to X-rays. In 2003 it became more variable, and it reached its greatest brightness in May 2005, when the flux in near ultraviolet was 4 times as great as that of M87's central energy source, the SMBH accretion disk. This represents a brightness increase at that wavelength of a factor of 90. The X-ray flux increased by a factor of 50, and similar, synchronized changes occurred at other wavelengths. The synchronization indicates that one mechanism is responsible for the variability at all wavelengths.

What the actual cause of the disturbance may be is not clear. Because of the great distance of HST-1 from the SMBH, its basic energy source must not be the central accretion disk itself. More likely HST-1 is a result of constriction of magnetic field lines, resulting in further acceleration of the particles making up the jet. Acceleration of charged particles causes radiation by the synchrotron process, and is evidenced by polarization of the emitted photons. Constriction of the jet may be a result of passage through a region of higher density of stars. The increased variability could mean that the jet has encountered a region of higher but varying stellar density. Alternatively, the jet may be passing through a patch of thick gas or dust, with excess radiation produced by the resulting particle collisions.

These results could explain the variability of light from other, more distant active galaxies, at least those which have strong jets, given that it's possible for a small region of the jet far from the SMBH to outshine the central source. However, another source of variability occurs when a jet is viewed at a very low angle to our line of sight, in which case any slight change of direction could cause an apparent change of brightness.

Refs:
• Hubble Space Telescope observations of an extraordinary flare in the M87 jet (3/6/09) – The Astronomical Journal
• Hubble Witnesses Spectacular Flaring in Gas Jet from M87's Black Hole (4/14/09) – Physorg.com (press release)
• Black Hole Creates Spectacular Light Show (4/14/09) – Space.com
• Black hole jet brightens mysteriously (4/15/09) – New Scientist
• Black hole spews out impressive light show (4/20/09) – Cosmos Magazine

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Pinwheel of Star Birth

Pinwheel of Star Birth (10/19/10)

This face-on spiral galaxy, called NGC 3982, is striking for its rich tapestry of star birth, along with its winding arms. The arms are lined with pink star-forming regions of glowing hydrogen, newborn blue star clusters, and obscuring dust lanes that provide the raw material for future generations of stars. The bright nucleus is home to an older population of stars, which grow ever more densely packed toward the center.

NGC 3982 is located about 68 million light-years away in the constellation Ursa Major. The galaxy spans about 30,000 light-years, one-third of the size of our Milky Way galaxy.

NGC 3982 – click for 984×1000 image

More: here

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Hey professor, I am honestly, truly sorry

Here are the (mostly disturbing) entries for the Final Exam Excuse Contest 2010. I have numbered them so that you can vote on your favorite in the poll at the end of the post. Some of these are real e-mails from students; some are not. Some are easy to detect as fakes; some are not.

This isn't the *nicest* way to end the academic term, but many of us have been getting these e-mails (or ones eerily similar..) in the past few weeks, so I hope the group-wallow in parody and sympathy will be emotionally satisfying in some way, or at least mildly entertaining.

My end-of-term experience has been supplemented by a cheating incident that makes me very sad because the students involved did not need to cheat (they were getting decent grades; now they are not), but I don't feel like writing about student "misconduct" right now.. maybe in 2011, when I am older and wiser and energized for the new year and term.

In the meantime, I am going on a blog-break while my family and I travel to an interesting part of the world. I shall likely return in early January.

Thanks for reading, and please vote in the poll at the end of this post.

**********

1.
Hey professor,

I am honestly, truly sorry for missing the final exam in your case. I am not usually so irresponsible and this is completely out of character for me. However, I feel that I should ask that you excuse my exam, in that one of my housemates was in the emergency room and I had to be there with him. You can check with the ER to prove that I did not just forget the exam. I have had some bad luck lately and I would never of missed the exam because your class is one of my favorites, but you know what they say: friends first. I am wondering if I would be able to make up the exam this Saturday, which is the only chance I have before my flight home for Christmas. I know you might want to take off points for the late exam, but I really am afraid about how it would affect my GPA, and I am a Senior. Thus, I could also do an extra credit project, if that would help you to give me the grade I really deserve were it not for missing the final. I will do any project you want, because I am very greatful to you for helping me even though I had my friend in the hospital. However, I might need to turn in my extra credit project after Christmas because I am going home and also my computer is at IT with a virus and I can't get my files off it. They said it would be done by now, so I have no idea when I'll get it. Is it ok if I can get an incomplete so that I can do my best work on the extra credit project? Also, I could take the final exam after Christmas so that I can really demonstrate my best work for this class, but I understand completely if you want me to take it on Saturday.

Thank you so much, I truly do love the class and I am sorry I let you down,

**********

2.

Dear Mr [prof]

i miss exam because due to car crash. i study all night and fall sleep drivering to school

i can bring police story to show why i missed. how can i get new exam and when

i must must must pass class or lose student visa!!!!!!!!!

thx Mr for helping me

**********
3.

hi fsp, im sorry i missed the final exam but yesterday morning my cat
was puking and i had to take her to the vet, and then after i got home
i was going to study but the cat puked again on my only clean sweater,
so i put the sweater and all my other dirty clothes in the laundry but
the dryer must have been 2 hot because my clothes came out all
munchkin-sized, so i went to the tj maxx to buy some new clothes but
then i left my lights on and the car wouldnt start. i tried to get my
friend to drive me to campus for the exam but he was all hungover from
celebrating the end of the semester and couldnt come pick me up at the
tj maxx, actually he couldnt get out of bed, and by this time i was
late for the exam and i would have called ur office but i lost my
phone. so its not my fault i wasnt there for the final yesterday but i
do really need to take the test because my dad said if i failed one
more class he would take away my cadillac escalade. i studied for a
whole hour and im sure i can get the a+ u told me i need to pass. i can
i can come in for the makeup exam tomorrow after 7pm or the next day
after 6 pls let me know which is best for u.

**********

4.

Dear Female Science Professor,

I regret to inform you that I could not make it to the final because
my grandmother died, and I need you to give me a make-up final so I
can get an A in this course. I don't expect you will, of course,
because you refused to give make-ups to all of my friends, whose
grandmothers also died to make them miss the exam. I don't know what
your big problem is. One day in class you made a big deal about
reading "real scholarly literature" instead of Wikipedia or stuff. So
here is this real paper for you
http://www.math.toronto.edu/mpugh/DeadGrandmother.pdf and it says, AND
I QUOTE: A student's grandmother is far more likely to die suddenly
just before the student takes an exam, than at any other time of year.
a student who is failing a class and has a final coming up is more
than 50 times more likely to lose a family member than an A student
not facing any exams. So I'm failing the class, and all my friends
are failing the class, and you don't think ahead to the fact that our
grandmothers were going to die? That dude says they worry themselves
to death. He says it is a mounting health epidemic growing with
exponential proportions that must be stopped or our society will
crumble down to its very foundations of society. So now I really
really have to get an A on the exam so I can go to med school to find
a cure to stop this SENSELESS LOSS OF LIFE. Please let me know what a
good time for the makeup would be. I work best from 1 am to 5 am so I
hope that works for you.

Sincerely,


Struggling Student

**********

5.

Professor XX,

My grandmother died today, or was it yesterday? I don't remember, it's
all a big emotional blur (/_\) . The funeral is on the day of the final
in a town far, far away and I have to go; she practically raised me
(well, on some weekends and holidays). Plus, her will stated that she
wanted me there specifically. Suffice it to say, I won't be able to make
it to the final exam. In fact, I'm so distraught that studying for any
exams is going to be almost impossible (we were really close).

I really need to pass this class (preferably with an A; my GPA is
hemorrhaging badly). I'm a super senior, and it's the last requirement
for my major. Also, it won't be offered again until Spring 2015! (Whose
great idea was that?!?!) In light of this, I propose that you average
the grades for my last 3 exams (with most of the weight on the highest
grade, of course), and use that as my final exam grade.

I know that you're probably thinking, "Yea, right. I want to see a death
certificate and an invitation to the funeral." Well, you're in luck! A
notarized death certificate and a notarized invitation to the funeral
will be available in a couple days.

Thanks for your cooperation (and the letter of recommendation ^_^ ) .

Very sincerely,

SuperSeniorIVX

**********
6.

Greetings Dr X,

I missed the midterm. I was miserably sick last week.

Sincerely,

**********

7.

Dear Prof X,

I am writing to explain that I accidentally gave my roommate, Student
Y, a pot brownie, so she was unable to take the final exam in your
class. Had I known about the exam, I would not have given this to her!

Please allow Student Y to make up the exam.

Sincerely,
Roommate Z

**********

8.

dear professor,

Last week I went to the clinic at the school to get my chest looked at
and I had whooping cough and broncitis, and I was given strong medication
for it. However, lastnight I took my medication on an empty stomach and this
morning I cant stop dry heaving and puking bile ( i am bulimic which also
makes my stomach sensitive). I need to go to the doctors or the hospital,
but I cannot drive right now. This is the second day that I have had this
reaction to my medication and it has cut into my study time and also put me
in pain. I dono what to do because I cannot miss this exam but I also cannot
lift my head out of the towilet for more than a few minutes. I donno how I
can write my exam like this. HELP, what should I do?

Student Y.

**********

9.

hi prof!
i just realized that I frgot 2 come to the final exam yesterday! I know the
exam was yesterday at 8 cuz I just checked on the finals calendr but between
my Chem midterm and my Psych paper it TOTALLY slipped my mind! could i
mebbe take the exam when I get back from break? Because I am actually
writing this on my iphone in the airport right now (also why the speling is
so bad - haha!). I'm super super sorry for the trouble!

Thanks soooooooooooooooo much in advance!!!
<3>

**********

10.

Hey professor,

I know it is late (2 AM the day of the final exam!!!!!!!) but I am supposed to
contact my professor in advance about taking a make-up exam for a good reason.
I recently found out that I have several final exams on the same day (today!!!) and so
according to university rules I get to take one as a make-up and I decided to take
yours later because it is my favorite class. The best time for me is Thursday at noon
so I will come to your office then and take the exam.
:)

**********

11.

Professor FSP,

I was starting to study but then I fell down on the floor and was sweating so I went to the clinic and they said I should rest and not study anymore, so that is why I can't take the final exam tomorow. My mom says I have to come home right away so she is picking me up at 2 pm which is when the exam starts. So the best thing to do is not to count this final exam in my grade. An Incomplete is NOT an option because I don't have time to make this up after the break. I calculated my grade as a B+ although I am really an A student and if you want to take into account my difficult circumstances and how hard I worked in your class an A would be good. Thanks for your help.

**********

12.

Dear Mr. [misspelled name]

I am in you're 1:00 class what meets in room 201 of the Maine building I rite you on a matter of grave concern
I had to miss the final exam what took place at 5PM in Maine 201 last tuesday because of a matter of vital importence; My mom cooked a really important dinner for me monday night and so do to the extreme difficulty of travel during the current season I had to go home during finals week because my mom insisted if you new my mom youd understand
I tryed to find you're office which you're web page says is MAine 202; but dint have any luck cuz the Maine building is to obscure and i couldnt find it so i couldnt find you're office and talk to you about it before
i wouldnt bother you about it except as its to important whereas my scholarship require that i keep a perfek 4.0 GPA thruout my intire collige years and so i need to make up the final I dont need you to work extra hard so its ok if you just give me the final that you gived everyone else as my friends said it wasnt to bad when they showed me the answers you passed out at the and
If you need confermation of the importence of the dinner you should contact my mom were in the phone book so were easy to find as our house is across the street from the collidge

you're devoted student John Smythe VII

**********

13.

Professor FSP, you're not going to believe this but just before
the final exam my pet python, Mimi, got loose somewhere in my
apartment building and the last time this happened my landlord
totally freaked and said if it happened again he would evict me
or kill my snake or both. So I had to look for her and I missed
the exam but the good news is that I found her and she was safe
but a little shook up and so then I had to get her calm and there
was no way I could email you until now. I am afraid to leave
her alone now but I could write a paper instead of doing the
final exam or I could take the exam after the break. Just let
me know which of these options you want to do.

**********


NOTE ADDED 12/21: Some late additions include:

14. I studied really hard last night and then slept through
the alarm clock

15. hey, i went 2 the room today and the normal time and no 1
was their. some kid was their and said you gave it during the
exam pd. anyways i work then so i couldn’t come.
i have time to make up the final after work tomorrow so i
can b their bout 4. thx

(no time to update the poll, but you could do a write in vote
in the comments)


Which one do you like the best?
1: ER excuse
2: car, police excuse
3: cat, laundry, friend etc. excuse
4: grandmother excuse, passive-aggressive
5: grandmother excuse. aggressive
6: pathetic but brief excuse
7: pot brownie excuse
8: detailed medical excuse
9: forgot exam excuse
10: last-minute excuse
11: illness, mom excuse
12: mom excuse
13: missing python excuse
  
pollcode.com free polls

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12 Months of Urban Science Adventures! © A Blogging Meme Story

All the cool kids other science bloggers were doing it, so I joined in.  Here is how it works:

Post the link and first sentence from the first blog entry for each month of the past year.

So here goes:

January: Setting My New Year's Resolutions

Happy New Year! I know I have been posting less than regularly, but I assure that it is a good sign of my progress on my dissertation

February: Wordless Wednesday: Still Stalking in the Snow

The SnowPocalypse isn't all bad.

March: Honey Bees Buzz with Individuality

I'm going to do something a little extra today - present a research paper to you.

April: It's April no fooling

Wow! It's April already.

May: My Final Commencement Ceremony

I graduate next week.

June: Preparing for summer camp

Oh, I love summer!

July: Blog Book Review: Jack in the Box presents Science Kids Mini-books

I love junk food. 

August: Election Day - Vote for Urban Science Adventures! (c)

It's Election day in many parts of the United States.

September: Making Science Make Sense with Dr. Mae Jemison

On Wednesday, September 8, 2010, THE Dr. Mae Jemison spoke a room of science, technology, engineering, and mathematics (STEM) educators and advocates at the state of Missouri STEM Summit presented by the Department of Elementary & Secondary Education.

October: A Wise Latina Scientist

We met in the Ladies Room near the main Auditorium at Bucknell University.

November: Snakes up close

I've got to make a confession.

December: Wordless Wednesday: Golden Life

The golden hues of autumn leaves make me smile.

It reads like a Mad Libs game. A little funny and choppy. 

But I am very happy about this year.  It's been the best, most blessed year of my life. 

Smooches to you all!

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Answering the Mail

In an attempt to be less of a slacker at answering my FSP e-mails, today I am going to give quick replies to some that have been lingering in my mailbox for varying lengths of time. That way, I will feel more psychically prepared for 2011. My apologies to those who e-mailed me but did not get a reply, including in this post.

Because I am providing only brief responses here, feel free to leave a comment expressing interest in a future discussion involving a more detailed and thoughtful consideration of topics raised in the e-mails.

(Some of the questions below have been edited for length)

Question: What is your take on giving information about other grad schools one is applying to? Some schools make it mandatory, some make it optional. What is the purpose of that and does it work in the student's favor to list all other schools?

Answer: How can this possibly be mandatory? Perhaps I am showing my ignorance, but I was aware only that some graduate programs request information on the "competition", if applicants are willing to provide it. Others can comment based on greater knowledge of this practice, but I can't see how it would work for or against an applicant to list these (or not). Just because you apply to a place doesn't mean you will be accepted. I think many places just want to look for overall trends in the data of applicant pools, not track where any particular applicant has applied, but others should correct me if their program uses this information in a different way.

**********

Question: Is having an open laptop during a talk disrespectful to a speaker and distracting to an audience, or are laptops a useful way of taking notes during the talk?

Answer: My personal opinion is that laptops or other electronic note-taking devices are acceptable during a talk. It can be distracting sitting next to someone who is type-type-typing throughout a talk, but if the typing is confined to jotting relevant notes or questions, I can deal with it. As a speaker, I assume that the open laptops are being used for a relevant purpose, although this assumption is clearly deeply flawed, as some people keep their eyes glued to their laptop screen throughout a talk, and occasionally laugh or smile at the screen at a time unrelated to anything in my talk that could be considered amusing (I think). That is rude.

**********

Question (aggregate of a number of e-mails with different situations but similar themes): What do you do if you are a tenure-track professor and many of your colleagues, including your department chair, are jerks?

Answer: My advice, which is not that awesome or satisfying but is the best I can suggest for those in this difficult situation, is to do your work as best you can, don't let the jerks get to you or destroy your enjoyment of your job if at all possible, find as many non-jerk colleagues as you can (within or beyond your department and including administrators), protect your students from the jerks, build a strong record of teaching/research/service, and get tenure. Then you can decide what to do: leave or stay. If you stay, you can then decide whether to confront the jerks or just do your own thing, or take the long view and become a leader in your department so you can change things. Just don't become one of them.

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Question: What do you do when a student writes to you, begging for a higher grade because [insert desperate reason]?

Answer: I can tell you what I do in these situations: I write back a short, sympathetic e-mail saying that I cannot and will not change the student's grade or give them an extra credit assignment. It is very sad when there is something important at stake for the student (e.g., a scholarship, financial aid, a chance to get into a desired program), but I try to forestall such unhappy events by giving students feedback throughout the course so that they know where they stand. I will help them if I can, before the final grade is determined, not after. The important thing is to be consistent and fair.

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Question: I have been told that one should decline to write a letter of recommendation if that letter is going to be really negative, or at least tell the person requesting the letter that the letter will be negative, and give the person a chance to find someone else (or take their chances with your unflattering letter). Actually, I have two questions: (1) Should one decline to write a negative letter (might this not be important information?) or inform the applicant that your letter will be negative, and if so, (2) How negative is negative? What if my letter would have both positive and some negative things in it? How do I decide whether I am being unfair to the letter-requester if I don't tell them how negative my letter will be?

Answer: This is a complex question. For now I will just say that I think it's fine to write a 'balanced' letter (code for 'has some positive and negative things in it'). Ideally, the person requesting the letter has some idea about what you think of him/her, although I know that is really hard to determine because there are all sorts of complex issues and anxieties involved in that. Nevertheless, if you have provided critical input to someone (e.g., about their research abilities, writing skills, productivity), given them a not-so-good grade in a class, or otherwise given them an open assessment of their accomplishments and/or abilities, there is nothing sneaky about writing some critical comments in a reference letter. Those reading the letter will appreciate an honest and frank assessment, and, as long as your negative comments are written in a professional way and you believe the criticisms to be fair, you've done what you were asked to do.

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Question: I am a graduate student and I have a 3-month old infant. I am doing fine, keeping up with my research, caring for my baby, and basically managing things as well as I can, although I feel that I am pretty much at the limit of what I can reasonably deal with. One of my office mates keeps complaining to me about how much work it is for him to care for his exotic reptile pet, which he fortunately keeps at home. Can I kill him?

Answer: I think you should seriously consider it and even make elaborate plans (in your head), but this might be something you want to put off until later, given that your plate is already full.

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There are some other good topics for discussion in my inbox, and I hope to get to them.. next year.

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Roots of unity and cyclotomic fields

In preparation for many good things that are to come, we need to have a talk about another important class of field extensions of ℚ – the cyclotomic extensions. (Check here for a list of previous articles on algebraic number theory.)

A cyclotomic field in general is a field that is an extension of some base field formed by adjoining all the roots of the polynomial f(x) = xⁿ-1=0 for some specific positive n∈ℤ to the base field. Usually, though not always, this will mean roots that lie in some large field in which f(x) splits completely and that contains ℚ as the base field, such as ℂ, the complex numbers. f(x) is known as the n^th cyclotomic polynomial. Mostly the same theory applies if the base field is a finite algebraic extension of ℚ, but we'll use ℚ as the base field for simplicity.

Since f(1)=0, x-1 is one factor of f(x), and f(x)/(x-1) = x^n-1 + … + x + 1 ∈ ℤ[x], with all coefficients equal to 1. If n is even, -1 is also a root of f(x). However, all other roots of f(x) in ℂ are complex numbers that are not in ℝ. Some of these roots, known as the "primitive" n^th roots of unity – denoted by ζ_n (or just ζ if the context is clear) – have the property that all other roots are a power ζ^k for some integer k, 1≤k<n. So the smallest subfield of ℂ that contains ℚ and all roots of f(x) is ℚ(&zeta), known as the n^th cyclotomic field.

It is possible to express all the roots of f(x) in the form e^2πi/n, where e^z is the complex-valued exponential function, which can be defined in various ways. The most straightforward way is in terms of an infinite series, e^z = Σ_0≤n<∞zⁿ/n!. The exponential function e^z can also be defined as the solution of the differential equation dF(z)/dz = F(z) with initial value F(1)=e, the base of the natural logarithms. So there is the rather unusual circumstance that the roots of an algebraic equation can be expressed as special values of a transcendental function. Mathematicians long hoped that other important examples like this could be found (a problem sometimes referred to as "Kronecker's Jugendtraum", a special case of Hilbert's twelfth problem), but that hope has mostly not been fulfilled.

The most well-known nontrivial root of unity is the fourth root, i=&radic(-1), which satisfies x⁴-1 = (x²+1)(x²-1) = 0.

All complex roots of unity have absolute value 1, i. e. |ζ|=1, since |ζ| is a positive real number such that |ζ|ⁿ=1. The set of all complex numbers with |z|=1 is simply the unit circle in the complex plane, since if z=x+iy, then |z|² = x²+y² = 1. (Note that the linguistic root of words like "circle", "cyclic", and "cyclotomic" is the Greek κύκλος (kuklos).) Since e^iθ = sin(θ) + i⋅cos(θ) for any θ, with θ=2πk/n the real and imaginary parts of a general n^th root of unity ζ=e^2πi⋅k/n are just Re(ζ)=sin(2πk/n) and Im(ζ)=i⋅cos(2πk/n).

There are many reasons why cyclotomic fields are important, and we'll eventually discuss a number of them. One simple reason is that roots of algebraic equations can sometimes be expressed in terms of real-valued roots (such as cube roots, d^1/3 for some d), and roots of unity. See, for example, this article, where we discussed the Galois group of the splitting field of f(x)=x³-2.

The set of all complex n^th roots of unity forms a group under multiplication, denoted by μ_n. This group is cyclic, of order n, generated by any primitive n^th roots of unity. (Any finite subgroup of the multiplicative group of a field is cyclic.) As such, it is isomorphic to the additive group ℤ_n = ℤ/nℤ, the group of integers modulo n. Because of this, many of the group properties of μ_n are just restatements of number theoretic properties of ℤ_n. For instance, each element of order n in μ_n is a generator of the whole group – one of the primitive n^th roots of unity. Since μ_n⊆ℚ(ζ_n), adjoining all of μ_n gives the same extension ℚ(ζ_n) = ℚ(μ_n).

Now, ℤ/nℤ is a ring, and its elements that are not divisors of zero are invertible, i. e they are units of the ring. They form a group under ring multiplication, which in this case is written as (ℤ/nℤ)^× (sometimes U_n for short). An integer m is invertible in ℤ/nℤ if and only if it is prime to n, i. e. (m,n)=1 (because of the Euclidean algorithm). The number of such distinct integers modulo n is a function of n, written φ(n). This number is important enough to have its own symbol, because it was studied by Euler as fundamental to the arithmetic of ℤ/nℤ. Thus φ(n) is also the order of the group (ℤ/nℤ)^×.

Let ζ=e^(2πi)m/n, for 0≤m<n, be an element of μ_n. The correspondence m↔e^(2πi)m/n establishes a group isomorphism between the additive cyclic group ℤ/nℤ and the multiplicative group μ_n. Modulo n, m generates ℤ/nℤ additively if and only if (m,n)=1, which is if and only if the corresponding ζ generates μ_n. So the number of generators of μ_n – which is the number of primitive n^th roots of unity – is the same as the order of (ℤ/nℤ)^×, i. e. φ(n).

One has to be careful, because the multiplicative structure of μ_n parallels the additive structure of ℤ/nℤ, not the multiplicative structure of (ℤ/nℤ)^×. (Because if ζ^M and ζ^N are typical elements of μ_n then ζ^M×ζ^N=ζ^M+N.) Hence even though there are φ(n) generators of μ_n, these generators do not form a group by themselves (a product of generators isn't in general a generator), so the set of them isn't isomorphic to (ℤ/nℤ)^×, even though the latter also has φ(n) elements. Give this a little thought if it seems confusing.

Moreover, the group (ℤ/nℤ)^× is not necessarily cyclic. It is cyclic if n is 1, 2, 4, p^e, or 2p^e for odd prime p, but not otherwise. Confusingly, if the group does happen to be cyclic then integers modulo n that generate the whole group are called "primitive roots" for the integer n. If (ℤ/nℤ)^× happens to be cyclic, then only those m∈(ℤ/nℤ)^× having order φ(n) are "primitive roots" that generate the group, while all m∈(ℤ/nℤ)^× have the property that if ζ∈μ_n has order n and generates the latter group, then so does ζ^m, as we showed above. Got that straight, now? This needs to be understood when working in detail with roots of unity.

Another reason for the importance of cyclotomic fields is that the Galois group of the extension [ℚ(ζ_n):ℚ] is especially easy to describe. Indeed, it is isomorphic to the group of order φ(n) we've just discussed: (ℤ/nℤ)^×. There's a little work in proving this isomorphism, but let's first note what it implies. Let G=G(ℚ(ζ)/ℚ) be the Galois group. It is an abelian group of order φ(n) since it's isomorphic to (ℤ/nℤ)^×. Further, any subgroup of G′ of G is abelian and by Galois theory determines an abelian extension (i. e., an extension that is Galois with an abelian Galois group) of ℚ as the fixed field of G′. Conversely, it can be shown (not easily) that every abelian extension of ℚ is contained in some cyclotomic field. (This is the Kronecker-Weber theorem.)

Half of the proof of the isomorphism is easy. Pick one generator ζ of μ_n, i. e. a primitive n^th root of unity. We'll see that it doesn't matter which of the φ(n) possibilities we use. Suppose σ∈G is an automorphism in the Galois group. Since σ is an automorphism and ζ generates the field extension, all we need to know is how σ acts on ζ. Since σ is an automorphism, σ(ζ) has the same order as ζ, so it's also a primitive n^th root of unity. Therefore &sigma(ζ) = ζ^m for some m, 1≤m<n. As we saw above, m is uniquely determined and has to be a unit of ℤ/nℤ, with (m,n)=1, in order for ζ^m to be, like ζ, a generator of the cyclic multiplicative group μ_n. Hence m∈(ℤ/nℤ)^×. Call this map from G to (ℤ/nℤ)^× j, so that σ(ζ)=ζ^j(σ). To see that it's a group homomorphism, suppose σ₁,σ₂∈G, with j(σ₁)=r, j(σ₂)=s. Then σ₂(σ₁(ζ)) = σ₂(ζ^r) = (ζ^s)^r = ζ^sr, hence j(σ₂σ₁) = j(σ₂)j(σ₁). j is clearly injective since j(σ)=1 means σ(ζ)=ζ, so σ is the identity element of G. Finally, to see that j doesn't depend on the choice of primitive n^th root of unity, suppose ζ^m with m∈(ℤ/nℤ)^× is another one. Then σ(ζ^m) = σ(ζ)^m = (ζ^j(σ))^m = (ζ^m)^j(σ).

Thus G is isomorphic to a subgroup of (ℤ/nℤ)^×. That's enough to show G is abelian, so the extension ℚ(ζ)/ℚ is abelian. To complete the proof of an isomorphism G≅(ℤ/nℤ)^× we would need to show that the injective homomorphism j is also surjective, i. e. every m∈(ℤ/nℤ)^× determines some σ∈G such that m=j(σ). We can certainly define a function from ℚ(ζ) to ℚ(ζ) by σ(ζ)=ζ^m for a generator ζ of the field ℚ(ζ). One might naively think that's enough, but the problem is that one has to show that σ is a field automorphism of ℚ(ζ).

The map σ defined that way certainly permutes the n^th roots of unity in μ_n, the roots of the polynomial f(x)=xⁿ-1. However, not all permutations of elements of μ_n, of which there are n!, yield automorphisms of ℚ(ζ). The problem here is that if z(x) is the minimal polynomial of some ζ, i. e. the irreducible polynomial of smallest degree in ℤ[x] such that z(ζ)=0, then by Galois theory the order |G| of the Galois group G is the degree of the field extension, which is the degree of z(x). Since G is isomorphic to a subgroup of the group (ℤ/nℤ)^×, and the latter has order φ(n), all we know is that |G| divides φ(n). It could be that other primitive n^th roots of unity have minimal polynomials in ℤ[x] that are not the same as z(x), though they have the same degree |G|. For σ to be an automorphism, σ(ζ) needs to have the same minimal polynomial as ζ, and we don't know that immediately from the relation σ(ζ)=ζ^m.

We will defer discussion of the rest of the proof that G(ℚ(μ_n)/ℚ)≅(ℤ/nℤ)^× for the next installment, since some new and important concepts will be introduced.

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Blogging my way to the North Pole.

Send me far, far away...only for a little while.  I'm trying it again*, entering a Quark Expedition Contest to Blog my Way to the North Pole, this time.  No thanks to climate change, the polar regions of our world are shrinking.  The Arctic regions, also known as the North Pole, are the most endangered of the two.
I would love the chance to be your daily blogging correspondent to show and tell you about this majestic ecosystem.

My appeal essay is titled: Following in Matthew Henson's footsteps.  If I win, then that's exactly what I will be doing, following in the footsteps of one of the world's greatest explorers, Matthew Henson.  He is known as the first African-American to reach the North Pole, along with 12 others including Robert Peary.

Mr. Henson. If I had Photoshop skills, I'd make a duplicate image next to him with my round face.  But can't you just see me bundled up like this.  Too cool! Literally and figuratively.

*Last year, I entered the Quark Expedition Contest to Blog my way to the South Pole. Thanks to great online supporters like Cynthia from Shimmy in My Spirit, Martin Lindsey from MartyBLOGs, the Blogging While Brown community, and my tweeps from Twitter. I had an amazing showing: 8th place out of 800 entries!

With your support, I could do equally as well again, if not better. Click here to vote.


Cynthia from Shimmy in My Spirit made this card for my Antarctic campaign. Cute, ain't it?

 I'll give Quark some props for revising the voting regime.  It's alot better than before.  You simply register with a valid email and click on the button to vote for me, and you can leave comments, now.  I really like that part. You can also vote for 4 additional people. The top 5 finalist are considered by a panel of judges and the winner they select will go on one amazing cruise.  I think this method is alot better and more fair than before. 

The voting continues until February 15, 2011 12 noon EST. The cruise expedition will take place June 23 - July 7, 2011 and includes visiting Helsinki, Finland and Murmansk, Russia before boarding a huge nuclear powered icebreaker cruise ship. Awesomeness!

So, please help me get there.  Vote for me and help me spread the news.  I appreciate it.

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Uninvited Speaker

Imagine this scenario:

You are organizing a conference session and thinking about possible invited speakers. You want a mix of old(er) superstars and dynamic early-career people.

One of the early-career candidates for an invited talk has told you that she plans to drop out of academia soon because she is moving to the city where her husband got a job. She might teach a bit but has no plans to continue as an active researcher, despite getting a PhD in a high-profile program and doing excellent and significant work.

Do you invite her anyway because her research is interesting, or do you give the invited talk slot to someone whose career prospects would benefit from the invitation and visibility?

Should a person's stated lack of interest in a future career involving research be a factor in this decision, assuming that there are other possible candidates whose research is as interesting and as compatible with the theme of the conference session? Or is the only thing that matters the research topic (and maybe also the individual's speaking ability)?

I think I would invite the quitting-research person anyway if she is clearly the best person for the session, no matter what her stated career goals. Even if the invited talk slot wouldn't benefit her career, it might benefit others in the audience (e.g., students or others who would learn something from her talk) or it might benefit the session overall to have a diverse group of invited speakers ("diverse" could refer to research topic, methods, career stage, gender etc.).

If, however, there were other excellent candidates who would give a similarly excellent and useful talk and who would also personally benefit from the invitation, I might well tilt towards inviting one of them instead.

One of my colleagues has been in this decision-making situation recently, so I was thinking about this type of scenario.

What would you do? (and why?)

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What Are You Saying?

Today's post, in response to a reader/professor who wrote to me when one of his international students complained that he (the professor) was hard to understand, is at Scientopia.

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Wordless Wednesday: Simply Red

 

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The Contest : 2010

Two years ago at about this time, some FSP readers submitted entries to the Statement of Purpose (SoP) contest. Last year, we did a Letter of Reference (LoR) contest. What shall it be this year?

I have contemplated various archetypal academic texts as the theme of the 2010 contest, but I eventually settled on this:

Write an e-mail message from a student to a professor explaining why you (the student) missed the final exam.

Although we all love and respect our students and appreciate the complexity of their lives, feel free to go a little crazy, make this a cathartic experience (if you need one), and convince me that there is or was no way you could make it to the final exam (but you need to pass this class and maybe even get an A).

It might be interesting if contest submissions are a mix of real e-mails and fabricated e-mails. I wonder if we will be able to tell the difference?

Send your submission to femalescienceprofessor@gmail.com, or leave it as a comment to this post and I will save it for when I post some or all of the results.

I am going to go on a blog-break next week, starting maybe Wednesday, but if you send me your submissions by Monday or Tuesday (12/21), I will try to get them compiled before I go off the grid.

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Sabbatical Spouses

My experience with sabbaticals and spouses involves planning with a professor-spouse, so I don't have any personal advice for those with non-professor-spouses. I do know that my colleagues with non-professor-spouses have done one or more of the following: (1) not gone on sabbatical at all; (2) taken a sabbatical, but of the "staycation"/don't-leave-home sort; or (3) taken a sabbatical consisting of short trips or somewhat extended stays at other institutions, but not for a long time (max 3 months, typically less). One friend took the kids to Europe for a few months and left the lawyer spouse at home (though he visited).

Some of my colleagues with professor spouses in different fields or institutions have taken separate sabbaticals at different times or in different places, but few people I know have chosen to do this.

For 2-professor families who want to coordinate a sabbatical, there are some Issues that typically arise, such as:

- You have to coordinate things so that you both apply for, and are granted, sabbaticals at the same time.
- You both have to write grant proposals to raise $ for the missing 1/2 salaries you both won't be receiving.
- You have to agree on a place to go.

Each of these is a potential pitfall. For example, I had to wait more than 10 years for my first sabbatical because my husband was a few years behind me in seniority, and then we had to get our department chair to agree that we could both have a sabbatical in the same year.

The third item in the list, however, consumes most of our sabbatical discussions.

Agreeing on a place to go involves many complex factors. Although my husband and I are in the same general field, we are in different subfields, and different institutions may or may not have interesting (or any) colleagues in one of our subfields. So first we have to figure out all possible places that could conceivably host both of us, given our research interests.

Then we discuss which of those places we actually want to go. Although by this point the list of possible places has been significantly reduced, especially if we add the further constraint that we prefer to spend our sabbaticals outside the US, one or both of us may have different preferences and priorities.

For our last sabbatical, there was a very obvious place that had outstanding colleagues and facilities, was in an interesting place, and that had colleagues who wanted to host us. Also, the institution had money to pay visiting scholars, and that was quite a nice bonus to an already appealing option. So we went there.

That was great, but what about the next sabbatical? We have been discussing this and have pretty much settled on a place we think we would both like to be, and we have ascertained that there are colleagues there who would like to have us around for all or part of a year.

How did we ascertain that? For our last sabbatical, we both knew people at this institution, and it was not at all awkward to discuss our hopes for a visit. In fact, I think one or both of us may even have been invited. For the next sabbatical, one of us was approached by a professor at the university about visiting, and the other started e-mailing colleagues (some current collaborators, others known only from their research) to see what they thought of the idea. They liked the idea.

Although it is a bit disconcerting to cold-email someone and ask "Would you host me for my next sabbatical?", there's nothing too scary about asking someone if you can have a desk and be an interactive member of their research group for a while. Also, making these requests gets easier as you get older and more egotistical, and you count on the fact that people will either be enthusiastic or will at least make up something reasonably nice to discourage you if they don't want to be your host.

Of course our daughter has been an important element in our sabbatical planning as well. She loved the sabbatical we took when she was in elementary school. It was an adventure, she learned a new language, and we did a lot of traveling. She missed her friends and cats, but she made new friends and we figured out where all the cats in the neighborhood lived, and went on frequent cat safaris to visit friendly felines. Not long ago we returned to our sabbatical city and followed our old cat safari route, and there was our favorite cat, sitting in her usual spot, as if she hadn't moved in years.

Now our daughter is looking forward to our next sabbatical in a different place, despite the disruptions it will cause to her schooling. Whether such disruptions are significant (so the sabbatical is not a realistic option) or not a big deal (so the sabbatical is worth it for all) will of course vary from family to family.

I am a big fan of sabbaticals for their recharging effects and for the opportunities they provide to meet and work with new people, live in a different culture, travel, think, and have fun. Even if I couldn't get enough grant money to replace my missing 1/2 salary, I would try to go anyway.

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Splitting of prime ideals in algebraic number fields

Our series of articles on algebraic number theory is back again. Maybe this time it won't be so sporadic. Stranger things have happened. The previous installment, of which this is a direct continuation, is here. All previous installments are listed here.

When we left off, we were talking about how to determine the way a prime ideal factors in the ring of integers of a quadratic extension of ℚ. Such a field is of the form ℚ(√d) for some square-free d∈ℤ. We were using very simple elementary reasoning with congruences, and we found a fairly simple rule, namely:

If p∈ℤ is an odd prime (i. e., not 2), and K=ℚ(√d) is a quadratic extension of ℚ (where d is not divisible by a square) then

1. p splits completely in K if and only if p∤d and d is a square modulo p.
2. p is prime (i. e. inert) in K if and only if d is not a square modulo p.
3. p is ramified in K if and only if p|d.

The prime 2 behaves a little more weirdly, but the result is that 2 ramifies if and only if d≡2 or 3 (mod 4); 2 is inert if and only if d≡5 (mod 8); 2 splits if and only if d≡1 (mod 8).

One limitation was that our simple reasoning made it necessary to assume that O_K, the ring of integers of K, was a PID (principal ideal domain).

Let's review what we were trying to do. We were investigating the factorization of a prime ideal (p)=pO_ℚ(√d) in O_ℚ(√d). If O_ℚ(√d) is a PID, then there is a simple approach to investigate how p splits. If p splits then (p)=P₁⋅P₂, where P_i=(α_i), i=1,2. Any quadratic extension is Galois, and the Galois group permutes the prime ideal factors of (p). The factors are conjugate, so if α₁=a+b√d we can assume α₂=α₁*=a-b√d. Hence (p)=(α₁)⋅(α₁*)= (α₁α₁*)= (a²-db²).

Taking norms (to eliminate possible units ε∈O_ℚ(√d)) reduces the problem to a Diophantine equation of the form ±p=a²-db². With the problem thus reduced, a necessary condition for (p) to split (or ramify) is that the equation can be solved for a,b∈ℤ. A sufficient condition to show that (p) is inert, i. e. doesn't split or ramify, is to show that the equation can't be solved.

Let's look at how that might work. For example, let d=3. Looking at the equations modulo 3, we have ±p≡a² (mod 3). That is, either p or -p is a square modulo 3. Say p=5. The only nonzero square mod 3 is 1, and 5≢1 (mod 3). However -5≡1 (mod 3), so could we have -5=a²-3b²? Suppose there were some a,b∈ℤ such that -5=a²-3b². Then instead of looking at the equation modulo 3, we could look at it modulo 5, and find that then a²≡3b² (mod 5). If 5 divides either a or b, it divides both, and so 25 divides a²-3b², which is impossible since 25∤5. Therefore 5∤b. ℤ/(5) is a field, so b must have an inverse c such that cb≡1 (mod 5). Therefore, (ac)² ≡ 3(bc)² ≡ 3 (mod 5), and so 3 is a square mod 5. But that can't be, since only 1 and 4 are squares modulo 5. The contradiction implies -5=a²-3b² has no solution for a,b∈Z.

All that does show 5 doesn't split or ramify in ℚ(√3), hence it must be intert, but this approach is messy and still requires knowing that the integers of ℚ(√3) form a PID. We need to find a better way. Fortunately, there is one. But first let's observe that this elementary discussion shows there is a fairly complicated interrelationship among:

1. Factorization of (prime) ideals in extension fields,
2. Whether a given ring of integers is a PID,
3. Whether an integer prime can be represented as the norm of an integer in an extension field,
4. Whether an integer can be represented by an expression of the form a²+db² for a,b∈Z (in the case of quadratic extensions),
5. Whether, for primes p,q∈Z, p is a square modulo q and/or q is a square modulo p.

The problem of representing an integer by an expression like a²+db² is a question of solving a Diophantine equation, and more specifically is of the type known as representing a number by the value of a quadratic form. This question was studied extensively by Gauss, who proved a remarkable and very important result, known as the law of quadratic reciprocity, which relates p being a square modulo q to q being a square modulo p, for primes p,q.

We will take up quadratic reciprocity soon (and eventually much more general "reciprocity laws"), but right now, let's attack head on the issue of determining how a prime of a base field splits in the ring of integers of an extension field. We will use abstract algebra instead of simple arithmetic to deal with this question. For simplicity, we'll assume here that the base field is ℚ, even though many results can be stated, and are often valid, for more arbitrary base fields.

Chinese Remainder Theorem

The first piece of abstract algebra we'll need is the Chinese Remainder Theorem (CRT). Although it's been known since antiquity to hold for the ring ℤ, generalizations are actually true for any commutative ring.

Let R be a commutative ring, and suppose you have a collection of ideals I_j, for j in some index set, j∈J. Suppose that the ideals are relatively prime in pairs. In general that means that I_i+I_j=R if i≠j, and further, the product of ideals, I_i⋅I_j, is I_i∩I_j when i≠j. If R is Dedekind, then each ideal has a unique factorization into prime ideals, and they are relatively prime if I_i and I_j have no prime ideal factors in common when i≠j. Let I be the product of all I_j for j∈J, which is also the intersection of all I_j for j∈J, since the ideals are coprime in pairs.

The direct product of rings R_i for 1≤i≤k is defined to be the set of all ordered k-tuples (r₁, ... ,r_k), for r_i∈R_i, with ring structure given by element-wise addition and multiplication. The direct product is written as R₁×...×R_k, or &Pi_1≤i≤kR_i.

Given all that, the CRT says the quotient ring R/I is isomorphic to the direct product of quotient rings &Pi_1≤i≤k(R/I_i) via the ring homomorphism f(x)=(x+I₁, ... ,x+I_k) for all x∈R.

The CRT is very straightforward, since f is obviously a surjective ring homomorphism, and the kernel is I, since it's the intersection of all I_i. (It's straightforward, at least, if you're used to concepts like "surjective" and "kernel".)

Now we'll apply the CRT in two different situations. First let R be the ring of integers O_K of a finite extension K/ℚ, and I_i=P_i, 1≤i≤g, be the set of all distinct prime ideals of O_K that divide (p)=pO_K for some prime p∈ℤ. Then (p)=P₁^e₁ ⋅⋅⋅ P_g^e_g, where e_i are the ramification indices of each prime factor of (p). An application of CRT then shows that O_K/(p) ≅ Π_1≤i≤g(O_K/P_i^e_i). Recall that for each i, O_K/P_i is isomorphic to the finite field F_qi, where q_i=p^f_i for some f_i, known as the degree of inertia of P_i. (This field is the extension of degree f_i of F_p=ℤ/pℤ.) Further, Σ_1≤i≤ge_if_i=[K:ℚ], the degree of the extension. Check here if you need to review these facts. Specifying how (p) splits in O_K amounts to determination of the P_i and the numbers e_i, f_i, and g.

The second situation where we apply CRT involves the ring of polynomials in one variable over the finite field F_p=ℤ/pℤ, denoted by F_p[x]. Let f(x) be a monic irreducible polynomial with integer coefficients, i. e. an element of ℤ[x]. Let f(x) be f(x) with all coefficients reduced modulo p, an element of F_p[x]. f(x) will not, in general, be irreducible in F_p[x], so it will be a product of powers of irreducible factors: Π_1≤i≤g(f_i(x)^e_i), where f_i(x)∈F_p[x]. Each quotient ring F_p[x]/(f_i(x)) is a finite field that is an extension of F_p of some degree f_i. In general, e_i, f_i, and g will be different, of course, from the same numbers in the preceding paragraph. But the CRT gives us an isomorphism F_p[x]/(f(x)) ≅ &Pi_1≤i≤g(F_p[x]/(f_i(x)^e_i)).

Now, here's the good news. For many field extensions K/ℚ, there exists an appropriate choice of f(x)∈ℤ[x] such that for most primes (depending on K and f(x)), the numbers e_i, f_i, and g will be the same for both applications of the CRT. Consequently, we will have O_K/(p) ≅ F_p[x]/(f(x)), because for corresponding factors of the direct product of rings, O_K/P_i^e_i ≅ F_p[x]/(f_i(x)^e_i). As it happens, most primes don't ramify for given choices of K and f(x), so that things are even simpler, since all e_i=1, and all factors of the direct products are fields.

We can't go into all of the details now as to how to choose f(x) and what the limitations on this result are. However, here are the basics. Any finite algebraic extension of ℚ (and indeed of any base field that is a finite algebraic extension of ℚ) can be generated by a single algebraic number θ: K=ℚ(θ), called a "primitive element". In fact, &theta can be chosen to be an integer of K. Then the ring of integers of K, O_K, is a finitely generated module over ℤ. (A module is like a vector space, except that all coefficients belong to a ring rather than a field.) The number of generators is the index [O_K:ℤ[θ]]. (ℤ[θ] is just all polynomials in θ with coefficients in ℤ.) If p∈ℤ is any prime that does not divide [O_K:ℤ[θ]], then the result of the preceding paragraph holds. If for some p and some choice of θ p does divide the index, then there may be another choice of θ for which p doesn't divide the index. Unfortunately, there are some fields (even of degree 3 over ℚ) where this isn't possible for some choices of p.

The situation is especially nice in the case of quadratic fields, K=ℚ(√d), square-free d∈ℤ. If d≢1 (mod 4); we can take θ=√d and f(x)=x²-d, since O_K=ℤ[√d]. If d≡1 (mod 4), then the index [O_K:ℤ[√d]]=2, and there's a possible problem only for p=2. However, we still have O_K/(p) ≅ F_p[x]/(x²-d) for all p≠2. From that it's obvious that, except for p=2, (p) ramifies if p|d, (p) splits if d is a square modulo p, or else (p) is inert. That is exactly the conclusion we began with at the beginning of this article, on the basis of elementary considerations. Only now we need not assume that O_K is a PID.

There are four important lessons to take away from this discussion.

First, there is a very close relationship between the arithmetic of algebraic number fields and the arithmetic of polynomials over a finite field. Not only do we have the isomorphism discussed above, but it turns out that a number of similar powerful theorems are true for both algebraic number fields and the field of quotients of polynomial rings over a finite field.

Second, a lot of the arithmetic of algebraic number fields can be analyzed in terms of what happens "locally" with the prime ideals of the ring of integers of the field.

Third, many of the results of algebraic number theory are fairly simple if the rings of integers are PIDs (or, equivalently, have unique factorization). Such results often remain true when the rings aren't PIDs, though they can be a lot harder to prove. Often the path to proving such results involves considering the degree to which a given ring of integers departs from being a PID.

Fourth, and perhaps most importantly, abstract algebra is a very powerful tool for understanding algebraic number fields – and it is much easier to work with and understand than trying to use "elementary" methods with explicit calculations involving polynomials and their roots.

We will see these lessons validated time and again as we get deeper into the subject.

So where do we go from here? There are a lot of directions we could take, so we'll probably jump around among a variety of topics.

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