Num­ber the­ory is a branch of math­e­mat­ics devoted pri­mar­ily to the study of the inte­gers, their addi­tive and mul­ti­plica­tive struc­tures and their prop­er­ties that set them apart from other rings (struc­tures with addi­tion and mul­ti­pli­ca­tion). Ques­tions that per­tain to the inte­gers (and gen­er­al­iza­tions of them) are said to be arithmetic.

Ele­men­tary num­ber the­ory refers to num­ber the­ory which does not use tools from alge­bra and analy­sis. Princeton’s course on this is MAT 214. Many math majors who are inter­ested in num­ber the­ory learn this mate­r­ial on their own instead of tak­ing the course.

In order to pro­ceed fur­ther, one must first study alge­bra (MAT 345, 346) and com­plex analy­sis (MAT335).These are essen­tial tools in ana­lytic num­ber the­ory and alge­braic num­ber the­ory. Ana­lytic num­ber the­ory (MAT 415) uses tools from analy­sis to study the inte­gers, and often is con­cerned with ques­tions about the asymp­tot­ics and dis­tri­b­u­tion of arith­metic data, like prime num­bers, class groups of num­ber fields, dis­crim­i­nants of num­ber fields, and so on. Com­plex analy­sis is espe­cially impor­tant, and in fact much of the sec­ond half of MAT 335 is con­cerned with the proof of the prime num­ber the­o­rem, one of the pio­neer­ing efforts in ana­lytic num­ber theory.

Alge­braic num­ber the­ory (MAT 415, MAT 447) uses alge­braic tech­niques to study num­ber fields, which are finite field exten­sions of the ratio­nal num­bers. Num­ber fields have very sim­i­lar prop­er­ties to the ratio­nal num­bers (they have a ring of inte­gers that behaves much like Z) but dif­fer from them in sub­tle ways that are of arith­metic inter­est. Unlike over the ratio­nal num­bers, unique fac­tor­iza­tion of inte­gers in a num­ber field as the prod­uct of prime num­bers does not always hold, but an anal­o­gous ver­sion for ideals does!

A third branch of num­ber the­ory is arith­metic geom­e­try, which is con­cerned with find­ing solu­tions of poly­no­mial equa­tions over fields that are not alge­braically closed, such as the ratio­nal num­bers. This sub­ject is closely related to alge­braic geom­e­try, which is the study of geo­met­ric objects defined by poly­no­mial equa­tions. With the rapid devel­op­ment of alge­braic geom­e­try over the last fifty years, arith­metic geom­e­try has become a very excit­ing field of study for today’s math­e­mati­cians. Alge­braic geom­e­try is usu­ally offered as MAT 416, which can be either a course on vari­eties or schemes and coho­mol­ogy, depend­ing on the pro­fes­sor. Com­mu­ta­tive alge­bra, taught in MAT 447, is a key tool in study­ing alge­braic num­ber the­ory and alge­braic geom­e­try, and it is a good idea to study it before doing too much alge­braic geom­e­try (the alge­braic num­ber the­ory course does not assume knowl­edge of it). Alge­braic geom­e­try is noto­ri­ously fright­en­ing, but is essen­tial for research in the math­e­mat­ics of today — so it’s best to learn it early!

Num­ber the­ory uses a sur­pris­ing amount of rep­re­sen­ta­tion the­ory, topol­ogy, dif­fer­en­tial geom­e­try, real analy­sis and com­bi­na­torics — in this field, more than any other, a broad base is cru­cial to be able to do research today.

A help­ful link [Show]A help­ful link [Hide]
Sam Ruth, a Prince­ton grad stu­dent, wrote a great arti­cle about learn­ing alge­braic num­ber the­ory, which goes into a lot of depth about what you should learn as an under­grad and grad stu­dent if you want to be a num­ber the­o­rist, and talks about some of the main ideas and the­o­rems in alge­braic num­ber the­ory, ana­lytic num­ber the­ory, and ellip­tic curves.
Here is the link.

Courses directly rel­e­vant to future num­ber the­o­rists [Show]Courses directly rel­e­vant to future num­ber the­o­rists [Hide]
MAT 214: Num­bers, Equa­tions, and Proofs
This is a class in ele­men­tary num­ber the­ory. The mate­r­ial in this class is impor­tant to know if you want to con­tinue in num­ber the­ory, and will some­times be assumed in MAT 419. This class is some­times taken instead of MAT 215 as an intro­duc­tion to proofs. Please see the math department’s page on this course for a descrip­tion.
MAT 419: Alge­braic Num­ber The­ory
This course uses a lot of alge­bra, so you should def­i­nitely have taken MAT 345 and MAT 346 first. Know­ing Galois the­ory is espe­cially impor­tant. A bit of com­plex analy­sis is used towards the end of the course, so hav­ing taken MAT 335 before­hand or con­cur­rently would be help­ful (there are even a few times where the same proof is done in both classes). There are prob­lem sets approx­i­mately every week, which are time con­sum­ing but extremely worth­while. The text for the class is Samuel, but as it is a very thin book, it can often be help­ful to look at other texts, such as Mar­cus. This course cov­ers quite a lot of mate­r­ial, includ­ing some class field the­ory.
MAT 454: Class Field The­ory
Class Field The­ory is about under­stand­ing the abelian exten­sions of a num­ber field. The major­ity of the course is spent prov­ing one very impor­tant the­o­rem. The proof has two main parts — one of which uses ana­lyt­i­cal meth­ods (from MAT 335), and the other of which uses group coho­mol­ogy (you should at least be com­fort­able with exact sequences before­hand), which is more alge­braic. MAT 419 leads into it quite nicely. Because its focus is on prov­ing one big result, this course tends to have less empha­sis on prob­lem sets. How­ever, you should plan on spend­ing a lot of time review­ing your notes, as the mate­r­ial is quite chal­leng­ing.
MAT 415: Ana­lytic Num­ber The­ory
This course will be taught in Fall 2012 for the first time in a while, so no addi­tional infor­ma­tion is known about this course. Please check the math department’s page and the registrar’s page for more infor­ma­tion.
MAT 447: Com­mu­ta­tive Alge­bra
This course will be taught in Fall 2012 dif­fer­ently than in the past, so no addi­tional infor­ma­tion is known about this course. Please check the math department’s page and the registrar’s page for more infor­ma­tion.
MAT 416: Intro­duc­tion to Alge­braic Geom­e­try (Kol­lar)
This is a course on vari­eties, which are sets of solu­tions to poly­no­mial equa­tions. Com­mu­ta­tive alge­bra is a pre­req­ui­site, either in the form of MAT 447 or by read­ing Atiyah and MacDonald’s clas­sic text and doing lots of exer­cises to get com­fort­able with the tools used in alge­braic geom­e­try. The course fol­lows Shafarevich’s text and focuses on aspects of vari­eties, their local and global geom­e­try, embed­dings into pro­jec­tive space, and the spe­cific case of curves which is extremely well-understood. The final third of the course con­sisted of stu­dent pre­sen­ta­tions about var­i­ous spe­cial top­ics like ellip­tic curves, sur­faces, res­o­lu­tions of sin­gu­lar­i­ties, alge­braic groups and oth­ers. This course is fast-paced and chal­leng­ing, but worth the effort. Home­works tended to vary in length, fre­quency and dif­fi­culty.
MAT 416: Intro­duc­tion to Alge­braic Geom­e­try (Katz)
This is a course on sheaves, schemes and the coho­mol­ogy of coher­ent sheaves on pro­jec­tive vari­eties. It fol­lows the well-known text by Hartshorne. Com­mu­ta­tive alge­bra is an absolute pre­req­ui­site and an intro­duc­tion to vari­eties is highly rec­om­mended since schemes and sheaves are very abstract objects and hav­ing a good stock of exam­ples in hand is vital to under­stand the mate­r­ial well. One unusual fea­ture of this course which adds to its dif­fi­culty is that the bulk of the mate­r­ial on schemes and sheaves is rel­e­gated to the read­ings and home­work while the instruc­tor lec­tures on the coho­mol­ogy of pro­jec­tive vari­eties. Thus in some sense, it is two courses rolled into one and one would be wise to treat it as such. This is arguably the most chal­leng­ing course offered by the math­e­mat­ics depart­ment due to the con­stantly steep learn­ing curve and the excep­tion­ally heavy work­load. A final expos­i­tory project makes for a fun fin­ish to what cer­tainly will be a gru­el­ing semes­ter.
MAT 455: Advanced Top­ics in Geom­e­try — Lie The­ory
The goal of this course is to study the struc­ture the­ory of Lie groups and Lie alge­bras. These objects are ubiq­ui­tous in math­e­mat­ics and are stud­ied using a vari­ety of alge­braic, ana­lytic and geo­met­ric tech­niques. This course cov­ers the geom­e­try, struc­ture the­ory, clas­si­fi­ca­tion and touches upon their rep­re­sen­ta­tion the­o­ries. Some back­ground in dif­fer­en­tial geom­e­try is essen­tial, mostly mate­r­ial from the first few weeks of MAT 355. Alter­na­tively, read­ing through the first few chap­ters of Spivak’s book on Dif­fer­en­tial Geom­e­try should suf­fice. A vari­ety of text­books are use­ful — in par­tic­u­lar, Adams, Humphreys and Bump.
MAT 424: Rep­re­sen­ta­tion The­ory (Bhargava/Zhang)
This course draws on many dif­fer­ent areas of math­e­mat­ics — alge­bra, geom­e­try and some com­bi­na­torics. The first half of the course focuses on study­ing the lin­ear rep­re­sen­ta­tions of finite groups — this is a com­pletely worked out the­ory and the course spent sub­stan­tial time work­ing out var­i­ous exam­ples of char­ac­ter tables of finite groups. The sec­ond half of the course focused on the rep­re­sen­ta­tion the­ory of Lie alge­bras, in par­tic­u­lar on the the­ory of high­est weights. The dif­fer­ence between MAT 455 and this course is that the pri­mary goal in this course is to clas­sify the irre­ducible rep­re­sen­ta­tions of the objects in ques­tion, while in MAT 455 the empha­sis is on their intrin­sic geom­e­try — hence, there is value to be derived in tak­ing both courses. The text­book fol­lowed is usu­ally Ful­ton and Har­ris. In case you are new to work­ing with ten­sor, sym­met­ric and wedge prod­ucts, spend some time before the course starts to become famil­iar with them. Of course, knowl­edge of basic group the­ory is essen­tial.
MAT 424: Local Fields (Katz)
This course focuses on the Galois the­ory of the p-adic num­bers, which are the canon­i­cal exam­ple of locally com­pact topo­log­i­cal field with respect to a non-discrete topol­ogy. It’s very impor­tant to know Galois the­ory very well before tak­ing this course, from MAT 346: Alge­bra II. After intro­duc­ing the key tech­niques and ideas of the p-adics, the instruc­tor devotes the rest of the semes­ter to dis­cussing var­i­ous spe­cial top­ics — a vari­ety of solu­tions to the inverse Galois prob­lem for S_n and A_n, Hilbert’s irre­ducibil­ity the­o­rem and oth­ers. The home­works are long but not dif­fi­cult and are very good at get­ting stu­dents com­fort­able with the style of argu­ment. There were no exams.

Con­tacts [Show]Con­tacts [Hide]
Ash­wath Rabindranath ’13 (arabindr[at]princeton[dot]edu)
Sarah Trebat-Leder ’13 (strebat[at]princeton[dot]edu)