Galois group

Definition

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 $p$.