# Field Arithmetic (10-M=VKAR)

## Winter Semester 2017/18

### Class: Mon, Wed 12:15-13:45 in SE 30

### Exercises: Tue 14:15-15:45 in SE 30

### Homework sheets: See German
version of this page

Field Arithmetic is the part of algebra which uses, among
other tools, Galois theory, group theory, and algebraic geometry in
order to study number theoretic and field theoretic questions. Finite
fields play a prominent role too. Typical examples which we will cover
in class are:
#### Advanced topics in Galois theory

Dedekind's modulo prime reduction criterion for Galois groups of
polynomials over the rationals, further methods for the computation
of Galois groups, formal Laurent series fields and local Galois groups.
#### Hilbert's Irreducibility Theorem

If *f(t,X)∈Q[t,X]* is an irreducible polynomial in the
variables *t* and *X* with rational coefficients,
then *f(a,X)* is still irreducible over the rationals for
infinitely many integers *a∈Z*.

### Literature

- M. Fried, M. Jarden:
*Field Arithmetic*, third edition, Springer.
- P. Müller: Ergänzungskapitel
von
*Einführung in die Algebra*.
- H. Völklein:
*Groups as Galois Groups, An Introduction*,
Cambridge University Press.

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