Zahlentheorie / Algebraic Number Theory - SS 2019

Instructor: Daniele Agostini.

Office hours: Monday, 11:00-12:00. Room 1.429. RUD 25.

Lectures: Monday 9:15-10:45, Room 1.013, RUD 25.
Monday 15:00-16:30, Room 3.006, RUD 25.

Exercise Sessions (Leonardo Lerer): Wednesday, 11:15 - 12:45, Room 1.013, RUD 25.

Exam: July 25, 10:00, Room 1.115, RUD 25. Registration open until July 11.
September 26, 10:00, Room 1.115, RUD 25. Registration open until September 12.

Exam rules: There are no formal requirements to access the exam, but students are strongly encouraged to try to solve the exercise sheets and to take active part in the lectures and the exercise sessions. In the exam, you are allowed to use your own notes, but not anything else such as books, smartphones, etc. You are free (and encouraged!) to use every result from the lectures, the exercises session and the exercise sheets, unless you are explicitly asked to prove one of them. For the exam, you will have 2 hours and 30 minutes of time.

Important announcements

The second exam will be on September 26 (and NOT on September 27!) at 10:00 in Room 1.115, RUD 25. More details on the exam rules can be found above. Results September 27 at 14:00 in my office.

Course description

The course will be a first introduction to Algebraic Number Theory, arguably one of the most important and beautiful subjects in Mathematics. In particular, we will study number fields and their rings of integers.

Prerequisites: basic abstract algebra and Galois theory. A previous knowledge of Commutative Algebra will be useful but not required.

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References : Notes for some of the lectures will appear below. We will not follow exactly any particular book, but the main inspirations for the course will be

• D. Marcus, Number fields. Springer.
• J. Milne, Algebraic Number Theory. Web notes.

There are many other beautiful references for Algebraic Number Theory. One of them, with a more advanced presentation than the references above, is:

• J. Neukirch, Algebraische Zahlentheorie. Springer.

Course log

A brief description of each lecture's content, together with some notes, is available here:

• 8 April 2019, 9-11. Presentation of the course. Fermat's theorem on primes that are sum of two squares. Notes.
• 10 April 2019, 11-13. Basics of Galois theory: algebraic extensions, Galois extensions, Galois group, fundamental theorem of Galois theory. Notes.
• 15 April 2019, 15-17. Quadratic extensions and cyclotomic extensions. Modules over commutative rings. (Leonardo).
• 24 April 2019, 11-13. Theorem of Hamilton-Cayley, integral extensions and basic properties, number fields and rings of integers. Quadratic integers. Notes.
• 29 April 2019, 9-11. Trace and norm. Equivalence of definitions. Trace is linear, norm is multiplicative, composition. Notes.
• 29 April 2019, 15-17. Trace is nondegenerate. Integral elements: norm characterizes units, finiteness of integral closure. Noetherian modules. Notes.
• 6 May 2019, 9-11. Finitely generated modules over PID. Torsion-free modules. Integral bases for number fields. Discriminant. Notes.
• 6 May 2019, 15-17. Formulas for the discriminant of a power basis. The discriminant of quadratic extensions. Squarefree discriminant. Local rings. Localization. Notes.
• 13 May 2019, 9-11. The localization of the integers at a prime ideal. Every nonzero ideal in a noetherian ring contains a product of nonzero primes. Nakayama's lemma. DVR. (Leonardo). Notes.
• 13 May 2019, 15-17. Structure theorem for DVRs: the maximal ideal is principal, every nonzero ideal is a power of the maximal ideal, discrete valuations. Dedekind domains. Localization of a Dedekind domain is a DVR. Notes.
• 20 May 2019, 9-11. Chinese Remainder Theorem. Unique Factorization Theorem for ideals in Dedekind domains. Every ideal in a Dedekind domain can be generated by two elements. Notes.
• 20 May 2019, 15-17. Contraction of prime ideals. The ring of integers of a number field is a Dedekind domain. Examples of unique factorization. Notes.
• 22 May 2019, 11-13. Exercise session: solution of Exercise 4.4 about properties of localization.
• 27 May 2019, 9-11. Splitting of prime ideals under extensions of Dedekind domains. Ramification index, inertia degree. The case of principal extensions. Notes.
• 27 May 2019, 15-17. Sum of ramification indexes multiplied by the inertia degrees is the degree of the field extension. The case of Galois extensions: the Galois group acts transitively on the primes lying over a fixed prime. Examples with quadratic extensions. Notes.
• 03 June 2019, 9-11. A prime ramifies in a number field if and only if it divides the discriminant . Examples. Fractional ideals. Notes.
• 03 June 2019, 15-17. The ideal group and the ideal class group of a Dedekind domain. The ideal norm. Notes.
• 17 June 2019, 9-11. The absolute norm. Minkowski's bound and applications: finiteness of ideal class number, examples. Notes.
• 17 June 2019, 15-17. Lattices: definitions, fundamental parallelepipod, volume. Minkowski's convex body theorem. Proof of Minkowski's bound. Notes.
• 24 June 2019, 9-11. Dirichlet's unit theorem: statement and examples with quadratic fields. The roots of unity in a number field. Notes.
• 24 June 2019, 15-17. Dirichlet's unit theorem: proof. Notes.
• 1 July 2019, 9-11. Pythagorean triples. Fermat's last theorem for n=4, regular primes, units in cyclotomic fields.Notes.
• 1 July 2019, 15-17. First case of Fermat's last theorem for regular primes. Notes.

Exercise sheets

If you want your exercise sheet to be corrected you should hand it in on Monday by 12:00, either after the morning lecture or during the office hours.

• Sheet 1: since Monday 22 April is Ostermontag, it should be handed in on Wednesday 24 April.
• Sheet 2: to be handed in on Monday, April 29.
• Sheet 3: to be handed in on Monday, May 6.
• Sheet 4: to be handed in on Monday, May 13.
• Sheet 5: to be handed in on Monday, May 20.
• Sheet 6: to be handed in on Monday, May 27.
• Sheet 7: to be handed in on Monday, June 3.
• Sheet 8: to be handed in on Monday, June 17.
• Sheet 9: to be handed in on Monday, June 24.
• Sheet 10: to be handed in on Monday, July 1.
• Sheet 11: this sheet will not be corrected, but it will be discussed in the lectures of July 8 and July 10.

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Last modified: Fri Sep 27 09:50:10 CEST 2019