Seminar: Algebraic Numbers, Galois Theory and Applications

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Generalities

Organizer: Lars Kindler
Date: Tuesdays, 4-6pm, Place: SR 140/A7, Arnimallee 7

Official website in course registry

Please send questions and comments to kindler - at - math.fu-berlin.de
or come to office 109, Arnimallee 3

Prerequisites: Linear Algebra and some familiarity with basic notions of algebra like groups, rings, fields, ideals, normal subgroups, etc.

Guidelines: This is a seminar, which means the participants give the talks. Please write to me if you would like to reserve one of the talks for you, or if you have any questions.

First meeting: Tuesday, April 18

Description

Algebraic numbers are complex numbers which are zeroes of a polynomial with rational coefficients. For example $\sqrt{2}, 2^{1/3}$, $i$ or $e^{\frac{2\pi i}{n}}$. In this seminar we want to study algebraic numbers in the context of modern algebra. The notion of an algebraic extension of fields is central. Galois theory describes all (separable) algebraic extensions of a given field using the language of group theory. We will first discuss the Galois theory for finite algebraic extensions. It depends on the prior knowledge of the participants of the seminar how extensive this first stage will be. We will then study several applications and generalizations.

You can find a comprehensive program here.

Talks


DateTopicSpeaker
18.04.2017Algebraic foundationsLars
25.04.2017Algebraic field extensionsLars
02.05.2017The algebraic closureJakub
09.05.2017SeparabilitySimon
16.05.2017Galois extensionsDaniel
23.05.2017Summary/InterludeLars
30.05.2017Galois correspondence for finite extensions Sophia
06.06.2017Examples David
13.06.2017Fundamental theorem of algebra Yannic
20.06.2017Norm and Trace Louis
27.06.2017Cyclic extensions Joaquim
04.07.2017Solvability by radicals Michael
11.07.2017Galois correspondence for infinite extensions
18.07.2017A little bit of Galois Cohomology/Kummer- and Artin-Schreier Theory