## Galois Group of Polynomial

Separable Polynomial
A polynomial over $F$ is said to be separable if it has no multiple roots (i.e., all its roots are distinct).

Galois Group of Polynomial
Let $f(x)$ be a separable polynomial over $F$. Let $K$ be the splitting field over $F$ of $f(x)$. Then the Galois group of $f(x)$ over $F$ is defined to be $\text{Gal}(K/F)$.