Dies ist die Homepage für die Vorlesung Algebraische Zahlentheorie I im Sommersemester 2018-2019 in Mainz.
Dozent: Ariyan Javanpeykar
Mittwoch 8h-10h, 04-422
Donnerstag 8h-10h, 04-422.
- Neukirch, Algebraische Zahlentheorie [Neu]
- Sharifi, Algebraic Number Theory [Sha]. Available online. Click here
- Samuel, Theorie algebrique des nombres
- Ireland, Rosen. A classical introduction to modern number theory.
- Jarvis, Algebraic Number Theory
- Lang, Algebraic number theory.
- Stewart and Tall, Algebraic Number Theory and Fermat’s Last Theorem.
- Weil, basic number theory.
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.
- Abgabe 10 Mai. Blatt 1 Click here
- Abgabe 24 Mai. Blatt 2 Click Here
- Abgabe 5 Juli. Blatt 3. Click here.
Plan der Vorlesung (roughly)
- Mi 17th April. Introduction. Statement of Dirichlet's unit theorem. Integral extensions. Section 1.2 of [Sh]
- Do 18th April. Integral extensions, normal rings. Section 1.2 of [Sha]
- Mi 24th April UFD's. Principal ideal domains. Noetherian rings. Dedekind domains.
- Do 25th April Dedekind domains. Trace and norm. Section 1.3 of [Sha]
- Mi 1st May NO LECTURE
- Do 2nd May Trace and norm continued. Section 1.3 and Section 1.4 of [Sha]
- Mi 8th May Fractional ideals, Dedekind domains vB
- Do 9th May vB Unique factorization of ideals in Dedekind domains Homework 1 deadline May 10th
- Mi 15th May vB Ideal class group and PID's
- Do 16th May vB Ideal class group
- Mi 22nd May RW DVR's
- Do 23rd May RW Orders Homework 2 deadline May 24th
- Mi 29th May Orders
- Do 30th May NO LECTURE
- Mi June 5th Ramification, I
- Do June 6th Ramification, II
- Mi June 12th Ramification, III
- Do June 13th NO LECTURE
- Mi June 19th Decomposition groups and inertia groups
- Do June 20th NO LECTURE
- Mi June 26th Cyclotomic fields, I
- Do June 27th Cyclotomic fields, II
- Mi July 3rd NO LECTURE
- Do July 4th NO LECTURE,
- Mi July 10th Cyclotomic fields, III
- 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
- Quadratic reciprocity