# Algebraische Zahlentheorie I, Sommer 2019

Dies ist die Homepage für die Vorlesung Algebraische Zahlentheorie I im Sommersemester 2018-2019 in Mainz.

Dozent: Ariyan Javanpeykar

Vorlesung:

Mittwoch 8h-10h, 04-422

Donnerstag 8h-10h, 04-422.

Literatur

1. Neukirch, Algebraische Zahlentheorie    [Neu]
2. Sharifi, Algebraic Number Theory [Sha]. Available online. Click here
3. Samuel, Theorie algebrique des nombres
4. Ireland, Rosen. A classical introduction to modern number theory.
5. Jarvis, Algebraic Number Theory
6. Lang, Algebraic number theory.
7. Stewart and Tall, Algebraic Number Theory and Fermat’s Last Theorem.
8. Weil, basic number theory.

Hausaufgaben

You are required to hand in homework by yourself. This will help you understand the material and follow the course as well as possible. I will personally e-mail you corrections and comments on your homework that should help you improve your writing, and also your understanding of the course material.

I recommend you write your solutions in LateX. This will increase the quality of your homework, and it's a great way to practice writing mathematics with LateX. To use LateX, download MikTeX and an editor (like TeXmaker or TeXworks). A good tutorial for using LateX can be found under this link.

I prefer you e-mail me your homework solutions. You can of course also just give them to me.

1. Abgabe 10 Mai.      Blatt 1  Click here
2. Abgabe 24 Mai. Blatt 2 Click Here
3. Abgabe 5 Juli. Blatt 3. Click here.

TAKE HOME EXAM: Abgabe 30 Juli.

Plan der Vorlesung (roughly)

1. Mi 17th April.  Introduction. Statement of Dirichlet's unit theorem. Integral extensions. Section 1.2 of [Sh]
2. Do 18th April. Integral extensions, normal rings. Section 1.2 of [Sha]
3. Mi 24th April  UFD's. Principal ideal domains. Noetherian rings. Dedekind domains.
4. Do 25th April Dedekind domains. Trace and norm. Section 1.3  of [Sha]
5. Mi 1st May NO LECTURE
6. Do 2nd May Trace and norm continued. Section 1.3 and Section 1.4 of [Sha]
7. Mi 8th May Fractional ideals, Dedekind domains vB
8. Do 9th May vB  Unique factorization of ideals in Dedekind domains Homework 1 deadline May 10th
9. Mi 15th May  vB  Ideal class group and PID's
10. Do 16th May vB  Ideal class group
11. Mi 22nd May RW  DVR's
12. Do 23rd May RW Orders Homework 2 deadline May 24th
13. Mi 29th May  Orders
14. Do 30th May NO LECTURE
15. Mi June 5th Ramification, I
16. Do June 6th Ramification, II
17. Mi June 12th Ramification, III
18. Do June 13th NO LECTURE
19. Mi June 19th Decomposition groups and inertia groups
20. Do June 20th NO LECTURE
21. Mi June 26th Cyclotomic fields, I
22. Do June 27th Cyclotomic fields, II
23. Mi July 3rd  NO LECTURE
24. Do July 4th NO LECTURE,
25. Mi July 10th Cyclotomic fields, III
26. Do July 11th. Quadratic reciprocity Homework 3 deadline July 5th, Take Home exam: deadline July 19th

Themen der Vorlesung (roughly)

• Basic notions in commutative algebra: Krull-dimension of a ring, integral  closure, normal rings, unique factorization domains, Dedekind domains, orders.
• Ramification theory.
• Cyclotomic fields