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).

• 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, Groups]]^U
• [[Notes - Galois Theory HT25, Group actions]]^U
• [[Notes - Galois Theory HT25, Fields and field extensions]]^U
• [[Notes - Galois Theory HT25, Galois groups and Galois extensions]]^U
• [[Notes - Galois Theory HT25, Bounds on the size of the Galois group]]^U
• [[Notes - Galois Theory HT25, Separability]]^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, I]]^?
• [[Problem Sheet - Galois Theory HT25, II]]^?
• [[Problem Sheet - Galois Theory HT25, III]]^?

To-Do List

Let $M = K(\alpha)$ be a simple algebraic extension of $K$. let $m _ \alpha \in K[x]$ be the minimal polynomial of $\alpha$ over $k$. then there is an isomorphism of $K$-extensions