Engelska

Varannan hösttermin

Galois Theory was originally developped by Évariste Galois to characterize the polynomial equations that may be solved with root extractions. It has later also be used to solve classical geometric problems, e.g to show that it is impossible to divide an angle into three equal parts using only a straight edge and a compass.
The course aims to provide a deeper understanding of field extensions, and a connection between the theory of polynomial equations and group theory.

The course treats:
Field extensions: splitting fields, normal extensions and separable extensions, field automorphisms, normal closures.Galois groups: Galois extensions, the Galois Correspondence, the Fundamental Theorem of Galois Theory.Polynomial equations: solvability by radicals, insolvable quintics, symmetric polynomials, cyclotomic extension.

be able to give a detailed account of the concepts, theorems and methods included in the course,

be able to identify the most important theorems in the course and present their proofs.

be able to, in connection with problem solving, integrate knowledge from different parts of the course,

be able to independently identify problems that can be solved by methods that are part of the course and use appropriate solution methods,

be able to explain the solution to a mathematical problem within the course framework, in speech and in writing, logically coherent and with adequate terminology.

be able to argue for the importance of Galois theory as a tool for solving problems in other areas of mathematics, such as the theory of polynomial equations.

Föreläsningar

Seminarier

Skriftlig tentamen

Muntlig tentamen

Underkänd, godkänd

FMAN10 Algebraic Structures.

Stewart, I.: Galois Theory. ISBN 1584883936.
Cox, D.A.: Galois Theory. Wiley, 2012. ISBN 9781118072059.
Rotman, J.: Galois Theory. Springer, 1990. ISBN 0387973052.

The book by Stewart is the main textbook. (Later editions exist.)
The other books are recommended additional reading.

FMAP25F

2025-01-14

Maria Sandsten