Field Theory
Algebraic Closure
Every field embeds into an algebraically closed field in which every non-constant polynomial has a root
Splitting Field
The smallest field extension over which a given polynomial factors into linear factors
Separable Extension
A field extension where every element's minimal polynomial has no repeated roots
Normal Extension
A field extension where every irreducible polynomial with one root in the extension splits completely
Primitive Element Theorem
Every finite separable field extension is generated by a single element
Finite Fields
For every prime p and positive integer n there is a unique field with p^n elements
Transcendence Basis
Every field extension has a transcendence basis: a maximal algebraically independent set over the base field
Frobenius Endomorphism
In characteristic p, the map x -> x^p is a ring homomorphism that generates the Galois group of finite fields