Number theory I.a - Algebraic number theory

Instructor: philippe.michel@epfl.ch

Assistant: filippo.berta@epfl.ch

Course: 
Wednesday 15:15–17:00 in MA A3 31.

On some (rare) occasions the course might be given via Zoom (and be recorded).

Exercices:
Wednesdays 17.15–19:00 in MA A3 31.

Suggested literature:
Pierre Samuel: Algebraic Theory of Numbers

Exceptionally there will be no course and no exercise session this week. The first course will take place on Sept. 17th and will start from the beginning by discussing basic yet important diophantine equations.

Fermat's last  theorem in the cubic case and the descent argument.

• Definition of Dedekind domains. 
• Unique factorization of ideals in Dedekind domains. 

• The Dedekind property for the integral closure in separable extensions (course recorded)
  

Relative Theory and first half of the degree formula

• Proof of degree formula in the general case. 
• Ramification criteria.

Relative Theory for Galois extension:

Transitivity of the action of Galois on the primes above a prime

Decomposition subgroup at a prime. Inertia subgroup.

Frobenius conjugacy classes

The Dedekind recipe II

• The canoncial isomorphism and Frobenius elements
• The Frobenius conjugacy class
• Application cyclotomic fields

• Geometry of numbers. Minkowski Theorems for lattices.
• Finiteness of the ideal class group of a number field.Towards bounds for the class number.
• Hermite's Theorem on finiteness of number fields of a given discriminant.
  

Dirichlet's unit theorem

• Class number formula

• End of proof of class number formula
• Dedekind zeta function class group L-functions and distribution of primes ideals in congruence classes

The exam of MATH-482 will take place on June 22, 16h15-19h15 Swiss Time.

This is a take-home exam in the form of a moodle assignement on this page (which will become accessible during the exam period).

INSTRUCTIONS

Regarding allowed documents and quotations

1  The exam is an unmonitored take-home exam. In particular there is no restriction on the documents you can access to during that period.

2. On the other hand you must do the exam ALONE with no kind of human, animal, vegetal or extraterrestrial help (in particular no collaboration with  other students of MATH-482 and no help from anyone else, including by asking questions on a forum etc...).

3. For the proofs you can directly use any result proven in the course (the pdf document) and you don't need to give the precise location within the pdf document or the course notes. You can also quote any (correct) result present in the statements of the exercises (and duly proven during the exercise sessions or in the solution written by Vignesh Nadarajan). 

4. On the other hand you CANNOT quote any part of a solution/proof: you can not write "we proceed like in the solution of Exercise XX of sheet YY". You have to write down the portion of the argument that you need. You cannot quote any other material either terrestrial or extraterrestrial (for instance you cannot say "Exercise XX is a special case of Proposition 1.1 of A. Wiles's article "Modular Elliptic Curves and Fermat's Last Theorem" Annals of Maths 1995")

5. You can freely assume and use any result stated in one part of the exam for another part (of course if both parts are in the same exercise, the assumed part must be anterior to the part to be proven !).

6. In case of doubt on the application of these instructions, I have the authorisation to organise a Zoom interview with you to have some clarification and discuss of your solution of specific exercise(s).

7. It is not necessary to have completed correctly all the questions of all the exercises to receive the maximal grade.

Uploading of the exam on Moodle

1. The length of the exam has been adjusted for an official duration of 2h15mins; the remaining 45mins is a "technical time" for you to upload a PDF copy of your exam on moodle).  Again, it is not necessary to have completed correctly all the questions of all the exercises to receive the maximal grade.

2. Your exam MUST be hand written (by yourself) on A4 white paper sheets (one side preferred but if you write on both sides you must make sure that the paper is thick enough to not be transparent);  another possibility is to use a tablet and a stylus (like an ipad with Goodnotes) with any handwriting recognition software turned off so that exam has your personal handwriting on it. You must write in blue or black ink. Use a new sheet when you pass to a new exercise.

3. You then have to scan the sheets to produce either one or several PDF files (one per exercise) as you prefer. Your NAME and SCIPER must be clearly present on the first page of each of the file(s); also indicate the total number of pages for each file.

4. You must therefore have uploaded the file(s) of your exam on the moodle of the course MATH-482 before June 22 2021 at 19h15 (Lausanne time). If you miss the deadline because of exceptional circumstances it will still be possible to upload the file until 20h15. However you have to contact me by email or phone to notify me of the reason of this situation.

5. In case of a problem/question during the exam period and after (until 20h15) you can reach me at  +41 79 535 0086.