MAT
6617

Théorie des Nombres/Number Theory

Automne/Fall 2014

Professeure/Professor:
Matilde
Lalín

Local/Classroom: Pav. André Aisendstadt 4186

Horaire/Class Times: mardi/Tuesdays 9h30-11h00, mercredi/Wednesdays 9h30-11h00

Bureau/Office:
Pav André Aisendstadt 5145

Disponsibilités/Office hours: mardi/Tuesdays 11h00-12h00, mercredi/Wednesdays 11h00-12h00

Tel:
(514) 343-6689

couriel/e-mail:
mlalin at dms . umontreal . ca

Manuel/Text: "Algebraic Number Theory, a Computational Approach", William Stein

Langage/Language: Anglais pour les présentations, français et anglais pour les autres activités/ English for lectures, French and English for everything else.

**Information:**

Devoir/Homework: (les dates et le numéro des devoirs sont sujets à changement/ due dates and number of homework assignments are subject to change)

- The solutions will be posted on 26 - novembre/November Non-devoir/Non-Homework 5 Solutionnaire/Solutions

- due/date limite: 12 - novembre/November Devoir/Homework 4 Solutionnaire/Solutions

- due/date limite: 29 - octobre/October Devoir/Homework 3 Solutionnaire/Solutions

- due/date limite: 8 - octobre/October Devoir/Homework 2 Solutionnaire/Solutions

- due/date limite: 24 - septembre/September Devoir/Homework 1 Solutionnaire/Solutions

Avis importants/Special Announcements:

- Dec-2: Solutions to the Final Exam

- Nov-26: I'll be holding office hours on Monday December 1, 1:30-3:30. If you see me any other time, you're welcome to ask me questions. Also, if you want to meet at any other time, feel free to e-mail me.

- Nov-17: The final exam will take place on December 2, 9:30-12:30, in the usual classroom. You can bring a cheat sheet, consisting of one letter-size paper (where you can write in both sides).

- Nov-12: If you need to see the solutions to the non-homework 5 before the date I plan to post them online, just send me an e-mail and I'll be happy to send them to you when you need them.

- Oct-29: L'examen final aura lieu le 2 décembre 9h30-12h30. The final exam will take place on December 2nd, 9:30-12:30

- Sept-10: Le devoir 1 est ici! Homework 1 is here!

- Les dates du devoir/Homework due dates: 24 septembre/September 24, 8 octobre/October 8, 29 octobre/October 29, 12 novembre/November 12

- Examen final/Final exam: le 2 décembre/ December 2, 9:30-12:30

- Devoir/Homework: 65%

- Examen final/Final exam: 35%

Thèmes/Topics:

- 3-Dec: (expected) Teaching evaluations (please come!), Dedekind zeta function, Dirichlet class number formula, statement of applications.

- 2-Dec: Exam! Usual classroom, 9:30-12:20 (also, office hours on Dec 1)

- 26-Nov: Examples of Frobenius, distribution of ideals (cases imaginary quadratic, and idea of the real quadratic, plus statement of the general case) (Marcus, chapter 6)

- 25-Nov: end of the proof that if p divides the discriminant, it ramifies, finite fields, Frobenius, Frobenius in cyclotomic extensions (Marcus, 108-114)

- 19-Nov: More about inertia and decomposition, inertia and
decomposition of 5 in
ℚ(√ 2 ,√ 5 , i) ,
totally split
(completely splits), totally
ramified, ramification
and discriminant (Marcus 102-105, half way proof of Thm 34 in page 112).
**This ends the topics to be included in the final exam.**

- 18-Nov: More properties of inertia and decomposition, example of ℚ(ω_23) (Marcus 101-102)

- 12-Nov: An example and some discussion of the Fundamental Theorem of Galois Theory, Galois theory applied to prime decomposition: Decompositon and inertia groups (Stein notes 104, Marcus 98-100)

- 11-Nov: Discrete subgroups of ℝ^n are lattices, the image of ᵩ is discrete, proof of Dirichlet's theorem (90-92 from the notes)

- 5-Nov: Examples of application for Mikowski's bound, introduction to Dirichlet unit theorem, the real quadratic case and fundamental units, relationship between units and norm, the function ᵩ, its kernel and image (notes, 84-89)

- 4-Nov: Proof that the class number is finite with an explicit compact convex symmetric set S, the discriminant is >1 for K a nontrivial extension of the rationals, some comments on the class number h (81-83 from notes and pages 139-140 from Marcus).

- 29-Oct: Lattices and Blichfeld's theorem, the lattice generated by the integers of K, the lattice generated by a fractional ideal (79-81 from the notes)

- 28-Oct: End of the proof of the Theorem of prime factorization in extensions (Marcus, Theorem 27 (1), page 80), real and complex embeddings, the class group and the Minkowski bound, statement of the finitness of the class group, application to ℚ(i) and ℚ(√ 10 ) (pages 77-79 from the notes)

- 15-Oct: How to factorize a prime in a number field extension, with the particular example of quadratic fields, (Marcus 74, 78-82, Theorems 25 and 27, we're missing the proof of Theorem 27 (1), notes, 4.2)

- 14-Oct: End of properties of the norm of an ideal, effect of embeddings on primes lying over a given prime, ramification and discriminant (Marcus 69-72)

- 8-Oct: Inertia degree, the norm of an ideal and ∑ fe=[L:K], informal definiton of Galois closure (Marcus, 64-69, excluding the proof of 22(c), chapter 6 of notes)

- 7-Oct: Application of CRT (I=(a,b), ℘^n/℘^{n+1} isomorphic to R/℘), factoring primes in extensions (primes of L lying over primes of K), examples of how 2, 3, and 5 factor in ℤ[i], definition of ramification index (rest of 5.2 from the notes, pages 62-64 Marcus)

- 1-Oct: Cancelation and division of ideals, every ideal is a product of prime ideals, gcd and lcm of two ideals, Chinese remainder theorem (rest of 3.1, and 5.1, and Lemma 5.2.2 from the notes, pages 59-61, Marcus)

- 30-Sept: Fractional and integral ideals, divisibility of ideals, every integral ideal contains a product of irreducible ideals, the fractional ideals form a group under multiplication (inverse of a fractional ideal), an introduction to ideal class group. (3.1 from the notes, pages 55-58 from Marcus, warning: I've mixed both sources)

- 24-Sept: Integral basis of KL, application to ℚ(ω_pq) (pages 33-35, Marcus), integrally closed, the ring of all algebraic integers, Dedekind domains, O_K is a Dedekind domain (beginning of 3.1 from the notes)

- 23-Sept: Discriminant of ω_p and the additive structure of a ring of integers, discriminant of a field, the ring of integers of ℚ(ω_p) is ℤ[ω_p] (pages 27-32, Marcus)

- 17-Sept: Discussion of norms, traces and embeddings of ℚ(∜ 2 ), Continuation of trace and norm (2.4 from the notes), discriminants: computation in terms of embeddings, trace, and discriminant of α (pages 24-27 Marcus)

- 16-Sept: Number fields, ring of integers, example of quadratic extensions (Corollary 2, page 15, Marcus), order, function fields (introduction and sketch of the relationship with algebraic curves), definition and ways to compute norm and trace (2.3, 2.4 from the notes)

- 10-Sept: Rings of Algebraic Integers, algebraic numbers, algebraic integers, minimal polynomial (2.3 from the notes), example of the cyclotomic polynomial (Thm3, page 17 Marcus)

- 9-Sept: Noetherian rings and modules -- ascending chain condition, short exact sequences, Hilbert Basis Theorem (2.2 from the notes)

- 3-Sept: ℤ[i] inside ℚ(i). Fermat equation.
Ideals
(principal, maximal, prime). Ideal factorization (example of 6
factorized into ideals in ℤ[√ -5 ]) and units.
Introduction to Noetherian
rings and modules, definitions of module, noetherian ring and module,
finitely generated, example of ℚ[x
_{1},x_{2}...] (2.2 from the notes)

- 2-Sept: Introduction to class. Integral Domains and divisibility. Unique Factorization Domains. ℤ[√ -5 ] (is not a UFD) and ℤ[i] (Gaussian integers, UFD). Primes in ℤ[i]. Pythagorean triples.

Ouvrages complémentaires/Other Resources:

- Number Fields, by D. A. Marcus. (I may use this book often.)

- Chapters 13 and 14 of Abstract Algebra, Dummit et Foote,3rd edition, Willey and Sons, 2004 (contains a good introduction to Galois Theory)

- Fields and Galois Theory by J. S. Milne.

- Algebraic Number Theory, by J. S. Milne.

- A Brief Guide to Algebraic Number Theory, by H.P.F. Swinnerton-Dyer.

- Algebraic Number Theory, by S. Lang.

- Algebraic Number Theory, by Neukirch.

- Dario Alpern's website with Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Dernière mise à jour/Last update: le 8 septembre 2014 (ou plus tard) / September 8, 2014 (or later)