Course - Galois Theory HT25


An introductory course on Galois Theory, focussed on building the theory to the point where we can prove the insolubility of quintics by radicals (i.e. there is no “quintic formula” like the “quadratic formula”). The core observation of Galois theory is a remarkable correspondence between field extensions (i.e. $\mathbb Q$ is a field, if you “add” $\sqrt 2$ as an element to $\mathbb Q$, you have the extension $\mathbb Q[\sqrt 2]/\mathbb Q$) and the so-called Galois group of the field extension, which is (in some specific way) the set of “symmetries” of the field extension.

This course actually answered some questions I had from studying Further Maths at A-level, like the relevance of the [[Further Maths - Roots of Polynomials]]A topic and why complex numbers $a + bi$ seemed very similar to something like $a + b\sqrt 2$ (they’re both vector spaces containing conjugation as a linear map).

Notes

The Part A [[Course - Rings and Modules HT24]]U is a prerequisite, especially:

The Prelims courses [[Course - Groups and Group Actions HT23]]U and [[Course - Groups and Group Actions TT23]]U are also very relevant:

And occasionally results from the linear algebra courses [[Course - Linear Algebra I MT22]]U, [[Course - Linear Algebra II HT23]]U and [[Course - Linear Algebra MT23]]U will be used.

Problem Sheets

To-Do List




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