MAS442   Galois Theory   (10 credits)

Description

Given a field K (as studied in MAS333/MAS438) one can consider the group G of isomorphisms from K to itself. In the cases of interest, this is a finite group, and there is a tight link (called the Galois correspondence) between the structure of G and the subfields of K. If K is generated over the rationals by the roots of a polynomial f(x), then G can be identified as a group of permutations of the set of roots. One can then use the Galois correspondence to help find formulae for the roots, generalising the standard formula for the roots of a quadratic. It turns out that this works whenever the degree of f(x) is less than five. However, the fifth symmetric group lacks certain group-theoretic properties that lie behind these formulae, so there is no analogous method for solving arbitrary quintic equations. The aim of this course is to explain this theory, which is strikingly rich and elegant.

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