The modern approach is to consider the automorphisms determined by these permutations. We note that any automorphism of a field F must leave its prime subfield fixed. An algebraic extension field F of K is called separable over K if the minimal polynomial of each element of F is separable. The field F is called perfect if every polynomial over F is separable. Any field of characteristic zero is perfect.
Any finite field is perfect.
The fundamental theorem of Galois theory 8.3.1. In the statement of the fundamental theorem we could have simply said that normal subgroups correspond to normal extensions. Thus E is a normal extension if and only if it is conjugate only to itself. In most results in this section we will assume that the fields have characteristic zero, in order to guarantee that no irreducible polynomial has multiple roots. When we say that a polynomial equation is solvable by radicals, we mean that the solutions can be obtained from the coefficients in a finite sequence of steps, each of which may involve addition, subtraction, multiplication, division, or taking n th roots.