MAT 6617


Théorie des nombres/Number Theory


Automne/Fall 2018

Professeure/Professor:    Matilde Lalín

Local/Classroom:    Pav. André Aisendstadt 4186

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

Bureau/Office:    Pav André Aisendstadt 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:    "Algebraic Number Theory, a Computational Approach", William Stein

"Number Fields" by D. A. Marcus.


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)

  • due/date limite: 3 - décembre/December
  • due/date limite: 19 - novembre/November
  • due/date limite: 5 - novembre/November
  • due/date limite: 15 - octobre/October


Avis importants/Special Announcements:

  • 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:

  • September 24 septembre (à venir/to come): definition and ways to compute norm and trace, (2.4 from the notes), ddiscriminants: computation in terms of embeddings, trace, and discriminant of α (pages 24-27 Marcus), the additive structure of a ring of integers.
  • September 18 septembre : Number fields, ring of integers, example of quadratic extensions (Corollary 2, page 15, Marcus), order, function fields, embeddings of ℚ(∜ 2 ), (2.3 from the notes)
  • September 17 septembre : algebraic integers, minimal polynomial (2.3 from the notes) example of the cyclotomic polynomial (Thm3, page 17 Marcus),
  • September 11 septembre : Hilbert Basis Theorem (2.2 from the notes), Rings of Algebraic Integers, algebraic numbers, (2.3 from the notes)
  • September 10 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[x1,x2...], ascending chain condition, short exact sequences (2.2 from the notes)
  • September 4 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)
  • A Brief Guide to Algebraic Number Theory, by H.P.F. Swinnerton-Dyer.
  • Algebraic Number Theory, by S. Lang.

  • Algebraic Number Theory, by Neukirch.



Dernière mise à jour/Last update: le 30 juillet 2018 (ou plus tard) / July 30th, 2018 (or later)