References: This video course in Galois Theory is based on Abstract Algebra by Dummit and Foote, and online lecture notes on Galois Theory of Milne [course notes]. The latter is my favourite and I have borrowed proof style and examples from Milne’s notes.
The videos lectures are linked via square brackets
We begin the course by introducing [irreducible polynomials] and four criteria (Rational Roots, Gauss theorem. mod p test, Eisentein’s Criterion) to determine irreducibility of the polynomial. Why do we even bother with irreducible polynomials ? This is because an irreducible polynomial for field is also maximal and prime.
In the [second lecture] we introduce field extensions which can be considered as a vector space over the base field. For example . Kronecker’s theorem ensures that we can always find an extension field in which will have a root. This extension field is constructed by simply adjoining the root of the polynomial to the base field. More formally we have . This is the key to Galois theory. If the polynomial has m roots. Then , the field cannot tell the roots apart.
Let us now introduce [ Algebraic Extensions], the key idea is to associate a minimal polynomial with each algebraic element. If is algebraic over then degree minimal polynomial and hence finite. This automatically leads to result that finite implies that the extension is algebraic.
In this lecture we prove that if [E:F] is algebraic and [L:E] is algebraic then [L:F] is algebraic. This lecture collects the techniques we have developed till now and uses them in a non-trivial way to prove an important result [Algebraic over Algebraic is Algebraic]
In this lecture we give the algebraic counterpart of [Geometric Constructions] using ruler and compass. A point (x,y) is constructible then where . This helps us prove the impossibility of duplicating a cube and squaring a circle
In the following post we introduce splitting fields. Splitting Fields