Galois group

Symbol-free definition
Let $$L/K$$ be an extension of fields that is Galois (viz algebraic, normal and separable).

The Galois group of this extension is defined as the group of field automorphisms of $$L$$ which fix every element inside $$K$$.

Realization
The question: does every finite group occur as the Galois group of some Galois extension is the famous inverse Galois problem. It has been proved that every solvable group occurs as the Galois group of some group.

A permutation representation
Consider a Galois extension of finite degree. Then, by the primitive element theorem, there exists a primitive element for the extension, or in other words, there exists a monic polynomial over $$K$$ such that the extension is generated by any root of $$K$$.

Now, every element of the Galois group acts as a permutation on the roots of this polynomial. In fact, the action is transitive (since the polynomial is irreducible) on the roots. This thus gives a transitive permutation representation of the Galois group.

Note, however, that the precise transitive permutation representation depends on the choice of the irreducible polynomial. Or does it?

A linear representation
Since the Galois group gives linear automorphisms of $$L$$ over $$K$$, it naturally gives rise to a linear representation.

Galois cohomology
Another rich source of representations (of all kinds) of the Galois group, is Galois cohomology.