MAT
6617

Théorie des nombres/Number Theory

Automne/Fall 2019

Professeure/Professor:
Matilde
Lalín

Local/Classroom: Pav. André Aisenstadt 4186

Horaire/Class Times: lundi/Mondays 13h30-15h30, mardi/Tuesdays 13h-14h

Bureau/Office:
Pav André Aisenstadt 5145

Disponsibilités/Office hours: mardi/Tuesdays 14h-15h, vendredi/Fridays 12h30-13h30

Tel:
(514) 343-6689

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

Manuel/Text: "Number Fields" by D. A. Marcus.

"Algebraic Number Theory, a Computational Approach", William Stein**Information:**

Devoir/Homework:

- due/date limite: 2 - décembre/December Devoir/Homework 5 solutionnaire/Solutions

- due/date limite: 18 - novembre/November Devoir/Homework 4 solutionnaire/Solutions

- due/date limite: 4 - novembre/November Devoir/Homework 3 solutionnaire/Solutions

- due/date limite: 7 - octobre/October Devoir/Homework 2 solutionnaire/Solutions

- due/date limite: 23 - septembre/September Devoir/Homework 1 solutionnaire/Solutions

Avis importants/Special Announcements:

- The teaching evaluations are available here since November 25. You are encouraged to evaluate this class! These evaluations are key to improving the teaching experience for everyone and are strictly confidential.

- Barème/Grade distribution: Devoir/Homework (100%) (Tous les devoirs seront réparties également.)/(Assignments will have the same weight.) Le devoir le moins bon de chaque étudiant sera ignoré. / The worst of the five assignment marks will be dropped.

Thèmes/Topics:

- December 3 décembre : Simon's presentation on p-adic numbers

- December 2 décembre : polar density, Dirichlet density, the density of the primes that split completely, the correspondence between Galois extensions and the sets of primes that split completely, Dirichlet class number formula, Dirichlet functions (Marcus 189- 192)

- November 26 novembre : Dedekind zeta function, (Marcus 182-188)

- November 25 novembre : Totally splits (completely splits), totally ramified, ramification and discriminant, independence of characters, Frobenius (Marcus 105-108, 112-114).

- November 19 novembre : class cancelled

- November 18 novembre : Decomposition and inertia groups, properties, example of ℚ(ω_23), inertia and decomposition of of 5 in ℚ(√ 2 ,√ 5 , i) (Marcus 98-105, Stein 104)

- November 12 novembre : Distribution of ideals in number field, the case of ℚ(√2) (Marcus 161-175), Galois theory appied to prime decomposition, introduction.

- November 11 novembre : proof of Dirichlet's theorem (90-92 from the notes), distribution of ideals in number fields, imaginary quadratic case (Marcus 158-161)

- November 5 novembre : relationship between units and norm, Discrete subgroups of ℝ^n are lattices, the image of ᵩ is discrete, (notes, 84-90)

- November 4 novembre : 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, introduction to Dirichlet unit theorem, the real quadratic case and fundamental units, (81-83 from notes and pages 139-140 from Marcus).

- October 29 octobre : Lattices and Blichfeld's theorem, the lattice generated by the integers of K, the lattice generated by a fractional ideals (80-81 notes)

- October 28 octobre : How to factor primes (proof) (80-82 Marcus), real and complex embeddings, the class group and the Minkowski bound, statement of finiteness of the class group, convex and symmetric about the origin sets, definition of lattice and fundamental parallelotope (77-80 notes)

- October 15 octobre : ramified primes divide the discriminant, how to factor primes (72-73, 78-80 Marcus).

- October 8 octobre : : factoring in extensions, the Galois case (69-71 Marcus)

- October 7 octobre : another application of CRT, factoring in extensions, ramification index, inertia degree, norms of ideals (pages 63-68 Marcus)

- October 1st/er octobre : gcd and lcm, Chinese Remainder Theorem, application to A=(a,b), UFD implies PID in Dedekind domains (rest of 3.1 and 5.1, and Lemma 5.2.3 from the notes, pages 59-62 Marcus)

- September 30 septembre : Dedekind domains, O_K is a Dedekind domain, 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), the equivalence relation for the group of fractional ideals, an introduction to ideal class group, prime decomposition (3.1 from the notes, pages 55-59 from Marcus, warning: I've mixed both sources)

- September 24 septembre : Integral basis of KL, application to ℚ(ω_pq) (pages 33-35, Marcus), O_K is noetherian and integrally closed, the ring of all algebraic integers, (beginning of 3.1 from the notes)

- September 23 septembre : discriminants: computation in terms of embeddings, trace, and discriminant of α (pages 24-27 Marcus), 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)

- September 17 septembre : order, function fields, embeddings of ℚ(∜ 2 ), (2.3 from the notes), definition and ways to compute norm and trace, (2.4 from the notes)

- September 16 septembre : algebraic integers, minimal polynomial (2.3 from the notes) example of the cyclotomic polynomial (Thm3, page 17 Marcus), Number fields, ring of integers, example of quadratic extensions (Corollary 2, page 15, Marcus), order

- September 10 septembre : short exact sequences, finitely generated modules over noetherian rings, quotients of noetherian rings, Hilbert Basis Theorem (2.2 from the notes), Rings of Algebraic Integers, algebraic numbers, (2.3 from the notes)

- September 9 septembre
Primes in ℤ[i]. 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}...], ascending chain condition, definition of short exact sequences (2.2 from the notes)

- September 3 septembre : Bienvenue à a la classe!/ Welcome to class! Quick definitions of integral domain, divisibility, units, irreducible, primes, unique factorization domain, ℤ[√ -5 ] (is not a UFD) and ℤ[i] (Gaussian integers, UFD). Statement of primes in ℤ[i].

Ouvrages complémentaires/Other Resources:

- 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 23 août 2019 (ou plus tard) / August 23rd, 2019 (or later)