Galois group

Definition without symbols
The Galois group of a field extension is the group of all field automorphisms on the extension that fix the inner field.

Definition with Symbols
Let $$L/K$$ be a field extension. The Galois group $$Aut(K/L)$$ is defined as the group of all automorphisms $$\sigma$$ on L such that $$\sigma(\epsilon)=\epsilon $$ $$\forall \epsilon \in K$$. When the field extension is Galois, then it is denoted as $$Gal(L/K)$$.

Facts

 * If the field extension is Galois then it must be Algebraic,Normal and Separable. By a result called the Primitive element theorem this guarantees that this field extension is primitive - i.e. there exists a polynomial $$p(x) \in K[x]$$ such that $$L/K = K/p(x) $$.
 * We can see that the Galois group must act on all the elements of the extension. So - since the extension is normal and separable it must act on the distinct roots of the polynomial $$p$$.We observe that the Galois group permutes the roots of the polynomial $$p$$.