Course  Galois Theory HT25
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 Lecture Notes
 Other courses this term: [[Courses HT25]]^{U}
 From the previous year:
 Course Webpage (previous year)
 Lecture Notes (previous year)
 Relevant textbooks:
Notes
 [[Notes  Galois Theory HT25, Fields and field extensions]]^{U}
 [[Notes  Galois Theory HT25, Group actions]]^{U}
 [[Notes  Galois Theory HT25, Bounds on the size of the Galois group]]^{U}
 [[Notes  Galois Theory HT25, Separability]]^{U}
 [[Notes  Galois Theory HT25, Galois extensions]]^{U}
 [[Notes  Galois Theory HT25, Main theorems of Galois theory]]^{U}
 [[Notes  Galois Theory HT25, Computing the Galois group]]^{U}
 [[Notes  Galois Theory HT25, Groups]]^{?}
 [[Notes  Galois Theory HT25, Solvable groups]]^{?}
 [[Notes  Galois Theory HT25, Solvability by radicals]]^{?}
 [[Notes  Galois Theory HT25, Determinant and discriminant]]^{?}
 [[Notes  Galois Theory HT25, Cubic equations]]^{?}
Related notes
The Part A [[Course  Rings and Modules HT24]]^{U} is a prerequisite, especially:
 [[Notes  Rings and Modules HT24, Basic definitions for rings]]^{U}
 [[Notes  Rings and Modules HT24, Ideals]]^{U}
 [[Notes  Rings and Modules HT24, Factorisation in polynomial rings]]^{U}
 [[Notes  Rings and Modules HT24, Factorisation]]^{U}
 [[Notes  Rings and Modules HT24, Fields]]^{U}
 [[Notes  Rings and Modules HT24, Polynomial rings]]^{U}
Also, the following notes from Prelims [[Course  Groups and Group Actions HT23]]^{U} and [[Course  Groups and Group Actions TT23]]^{U}:
 [[Notes  Groups HT23, Normal subgroups]]^{U}
 [[Notes  Groups TT23, Group actions]]^{U}
 [[Notes  Groups TT23, Orbits and stabilisers]]^{U}
 [[Notes  Groups TT23, Quotient groups]]^{U}
Anki filter:
"Breadcrumb:*Galois*" OR "Breadcrumb:*Basic definitions for rings*" OR Breadcrumb:*Ideals* OR "Breadcrumb:*Modules HT24, Factorisation*" OR Breadcrumb:*Fields* OR "Breadcrumb:*Polynomial rings*" OR "Breadcrumb:*Normal subgroups*" OR "Breadcrumb:*Group actions*" OR "Breadcrumb:*Orbits and stabilisers*" OR "Breadcrumb:*Quotient groups*"
Problem Sheets
 Sheet 1, partial solutions
 From the previous year:
Lectures
ToDo List

Why care about an upper bound on the size of the Galois group?
 Because, paired with another result, it lets you conclude that $ \vert \text{Gal}(K/F) \vert = [K : F]$ is separable
 Add the example from Sheet 1 where there is a polynomial over a field with a positive characteristic that is not separable (every polynomial over a field of characteristic zero is separable in our definition of separability).
 What results from the first half of the course depend on $F$ being a field with characteristic zero?
 Prove Lemma 4.20 when I understand why it is useful
 Example of $F \subseteq L \subseteq K$ where $K/L$ is Galois but $L/F$ is not Galois
 Example calculations of the Galois group of field extensions, would be good as a separate entry full of examples
 Would be a good idea to be a proper refresher on normal subgroups
 Why aren’t the automorphisms just all permutations of the roots? An example of a field and an “automorphism” where this wouldn’t work?
 Read Chapter 10, 13, 14 of [[Abstract Algebra, Judson]]^{N}
 Proof and significance of Corollary 6.7?

Theorem 6.8 seems important, it would be good to:
 Add a diagram
 Add an intuitive explanation of the proof
 Understand why we cannot immediately apply the inductive hypothesis
 Online tool for visualising Galois correspondence?
 Better understanding of proposition 6.9 and a counterexample in the case of nonzero characteristic.
 What exactly do the commutative diagrams represent?
 Proof of proposition 6.17
 Proof of proposition 6.18
 Details for cubics and quartics