Galois Theorem

Question 1

radical extension

Answer 1

there is an ascending chain of simple extensions K<K(a1)<…<K(a1,…,an)=F
such that for i there exists pi such that ai^pi in K(a1,…,ai-1)

Question 2

solvable by radicals over K

Answer 2

there exists radical extension K<L such that K<F<L where F is the splitting field of f over K

Question 3

solvable group

Answer 3

ascending chain of subgroups id<…<G for which Hi is normal in Hi+1 and all the factors Hi/Hi-1 are abelian

Question 4

Galois group of f over K

Answer 4

Galois group of K<F where F is the splitting field of f over K

Question 5

Galois Theorem

Answer 5

The polynomial f in K[x] is solvable by radicals over K if and only if the Galois group of f over K is solvable