MAT
6617

Théorie des Nombres/Number Theory

Automne 2010

Professeur/Professor:
Matilde
Lalín

Local/Classroom: 5183 André Aisendstadt

Horaire/Class Times: Lundi/Mondays 15:30-16:30, Mercredi/Wednesdays 14:30-16:30

Bureau/Office:
5145 André Aisendstadt

Disponsibilité/ Office hours: Lundi/Mondays 14:30-15:30, Mercredi/Wednesdays 13:30-14:30

Tel:
(514) 343-6689

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

Ouvrage/Text: "Number Fields", by D. A. Marcus

Langage/Language: Anglais/English

**Information:**

Devoir/Homework:

- due 20 - Sept Devoir/Homework 1-Réponses/Solutions

- due 4 - Oct Devoir/Homework 2-Réponses/Solutions

- due 18 - Oct Devoir/Homework 3-Réponses/Solutions

- due 3 - Nov Devoir/Homework 4-Réponses/Solutions

- due 15 - Nov Devoir/Homework 5-Réponses/Solutions

- due 29 - Nov Devoir/Homework 6-Réponses/Solutions

Avis importants/Special Announcements:

- 13 Oct: On the week of Oct 18-20, we will start talking about splitting in Galois extensions. Please review the Fundamental Theorem of Galois Theory if you are not familiar with it.

- 21 Sep: From now on, if I ask for n homework problems and you submit m>n, I will choose n out of those m and grade only those. This does not apply to the grading of homework 1. In homework 1, I am only grading more problems for the people who have a chance to improve their mark: i.e., for those of you who have 3 perfect problems, I am not grading more than that. If you want me to take a look at a problem that I haven't grade, you are welcome to bring it to my office hours.

- 14 Sep: A good Galois theory reference: Fields and Galois Theory by J. S. Milne. For a crash course, look at pages 1-12 and 17-32.

- See Homework due dates

Thèmes/Topics:

- 31-11: (tentative) Dirichlet class number theorem, L functions, Chap 8, some idea. This is the final class!

- 29-11:Continuation of Dedekind zeta function, polar density and some of its consequences.

- 24-11: Continuation of distribution of ideal classes (sketch of the general case and focus on the real quadratic case). Dedekind zeta function: definition and convergence over Re(s)> 1-1/[K:Q], with emphasis on the particular case of the Riemann zeta function.

- 22-11:Distribution of ideal classes (proof of the imaginary quadratic case, sketch of the general case).

- 17-11: The unit theorem.

- 15-11: End of computation of Minkowski bound, intro to the unit theorem, Class surveys.

- 10-11: Lattices and their volumes, image of O_K is a lattice, Minkowksi's theorem, computation of Minkowski bound (we're still missing the volume V_r,s(n) in the book's notation)

- 08-11: Class number is finite, number of real and complex embeddings, statement of the bound with Minkowski bound

- 03-11: p ramifies iff it divides the discriminant. Frobenius.

- 01-11: Splits completely and unramified in composite of extensions and in the normal closure.

- 20-10: Continuation of thm 28 and its consequences for normal extensions. Intermediate fields.

- 18-10: End of splitting in general extensions, splitting in Galois extensions, decomposition and inertia. Halfway through thm 28, understanding the relation between decomposition and inertia for a single prime.

- 13-10: n=efr for normal extensions, ramification and the role of the discriminant, how to find splitting when things are nice (we did only half the proof, point 1 from theorem 27 of Marcus, chap 3), splitting in quadratic extensions.

- 6-10: Proof that sum_i e_i f_i =n, and proferties of the number of elements of the quotient ring. Intro to spliting in normal extensions. The Galois group acts transitively in the primes lying over P.

- 4-10: Spliting of primes in extensions, ramification index and inertia degree.

- 29-9: Unique factorization in prime ideals, a few words about how to prove that these ideals are prime (in an example), Chinese remainder theorem (from Appendix 1 of book), GCD, LCM of ideals, UFD implies PID in Dedekind domains

- 27-9: Number rings are Dedekind, every proper ideal I in a Dedekind domain R contains a product of primes, There is a gamma in K - R such that gamma I is included in R, There is a J such that IJ principal. Definition of ideal class group.

- 22-9: The ring of integers of Q(w_p). The ring of integers of the composite of two number fields. Entering chapter 3 of book! Intro about ideals in commutative rings with unity, Noetherian rings, definition of Dedekind domain. Statement that every number ring is Dedekind.

- 20-9: Discriminant of omega_p, the ring of integers is a free abelian group of rank r, definition of integral basis, integral basis have the same discriminant, this defines discriminant of a number field, if the disciminant of r integral elements is square-free, then we have an integral basis

- 15-9: End of proof of the composition of Trace and Norm in extension towers, discriminants (definition as square of determinant of sigma(alpha), proof that is the same as the determinant of the Trace matrix, formula for disc(alpha) in terms of the norm of the derivative of the minimal polynomial)

- 13-9: Continuation of cyclotomic fields (characterization of the Galois group), traces and norms (definition, application to determine units, irreducibles)

- 8-9: Algebraic integers, number fields, number rings, the proof that number rings are rings, examples of number fields (quadratic, cyclotomic), the number rings of quadratic fields, begining of cyclotomic fields (description of the roots of the cyclotomic polynomial) (chapter 2 of Marcus)

- 1-9: Introduction to class. Discussion about Pythagorean triples and Fermat equation (chapter 1 of Marcus without the discussion of regular primes), some basic facts about divisibility in integral domains (definition of primes, irreducibles, units, UFD), example of Gaussian integers and Z[sqrt{-5}] (6 has two factorizations in irreducibles in this ring) discussion of norms for those cases.

Ouvrages complémentaires/Other Resources:

- 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.

- A Brief Introduction to Classical and Adelic Algebraic Number Theory, by William Stein.

- 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: 29 septembre, 2010 (ou plus tard) Last update: September 29, 2010 (or later)

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