Galois Groups, Field Extensions, and Radical Solvability

All flashcards

What is a Galois group of a field extension $L/K$?

The group of automorphisms of $L$ that fix every element of the base field $K$.

What is a splitting field of a polynomial over $K$?

The smallest field containing $K$ and all roots of the polynomial.

What does the fundamental theorem of Galois theory relate for a finite Galois extension?

Intermediate fields and subgroups of the Galois group, with inclusion reversed.

What is a solvable finite group?

A group with a series of normal subgroups whose successive quotients are cyclic of prime order.

What does the inverse Galois problem ask?

Which groups can occur as Galois groups over a specified base field.

Quiz questions

1. For a finite Galois extension $L/K$, let $E$ be an intermediate field and let $H$ be the subgroup fixing every element of $E. Which degree relation holds?Answer: $[E:K]=[G:H]$, where $G=\operatorname{Gal}(L/K)$.
2. What is the splitting field of a polynomial over a field $K$?Answer: The smallest field containing $K$ and all roots of the polynomial.
3. Over a field of characteristic zero, what group-theoretic condition characterizes a polynomial's solvability by radicals?Answer: Its Galois group is solvable.
4. What is known about realizing finite groups as Galois groups over $\mathbb{Q}$?Answer: Every finite solvable group is known to occur, but the general question remains open.
5. What kind of field-extension tower corresponds to compass-and-straightedge constructible lengths?Answer: A tower built by successive extensions of degree two.