Course - Galois Theory HT25

• Course Webpage
• Lecture Notes
• Other courses this term: [[Courses HT25]]^U
• From the previous year:
  • Course Webpage (previous year)
  • Lecture Notes (previous year)
• Relevant textbooks:
  • [[Abstract Algebra, Judson]]^N
  • [[Algebra, Artin]]^N

Notes

• [[Notes - Galois Theory HT25, Fields and field extensions]]^U
• [[Notes - Galois Theory HT25, Groups]]^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, Solvable groups]]^U
• [[Notes - Galois Theory HT25, Solvability by radicals]]^U
• [[Notes - Galois Theory HT25, Kummer extensions]]^U
• [[Notes - Galois Theory HT25, Determinant and discriminant]]^U
• [[Notes - Galois Theory HT25, Cubic equations]]^U
• [[Notes - Galois Theory HT25, Quartic equations]]^?
• [[Notes - Galois Theory HT25, Quintic equations]]^? (?, sort of not in the notes but probably useful to cover, lots of detail in [[Algebra, Artin]]^N)
• [[Notes - Galois Theory HT25, Finite fields]]^U
• [[Notes - Galois Theory HT25, Cyclotomic extensions]]^U

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:
  • Sheet 1, partial solutions
  • Sheet 2
  • Sheet 3, partial solutions
  • Sheet 4, partial solutions
• [[Problem Sheet - Galois Theory HT25, III]]^?

Lectures

To-Do 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, especially quartics (but also especially cubics).
• Relevance of $\Gamma _ n$ acting faithfully on $\mu _ n(\mathbb C)$ by group automorphisms
• Lemma 7.16
• Is there an alternative proof that the nth cyclotomic polynomial is irreducible? There is a much shorter proof for the prime case in [[Notes - Rings and Modules HT24, Factorisation in polynomial rings]]^U.
• Semidirect products
• What does it mean for a subgroup to be transitive?