Abstract algebra is the study of the algebraic structures for themselves. Its beginning is the development of the **Galois theory** by Évariste Galois. Since then the focus in on **groups**, **rings** and **fields**.

## Group theory

- Fundamental theorem on homomorphisms e1
- Fermat’s little theorem
- Existence of normal subgroups in p-groups
- Every finite p-group is solvable
- The direct product of a group
- There is only one group of order 15 (Sylow)

## Ring theory

- Finite Ring e1
- Units and zero divisors
- Ring embedded in an endomorphism ring
- x^3+x^2+x+2 is irreducible in the integers
- In a boolean ring every prime ideal is a maximal ideal

## Field theory

- An infinite field extension has an infinite number of intermediate fields
- Roots and multiplicity of the polynomial x
^{3}– 2 - Roots and multiplicity of the polynomial x
^{5}– x - Every finite integral domain is a field
- Q(i) and Q(√2) are isomorphic as vector spaces, but not as fields
- Field extension and multiplicativity formula for degrees