Solvable Group: Need Help Understanding Fraleigh's Exercise

[emptyboat]

Hello...

I have a question about the solvable group.

I read a Fraleigh's 'A first course in abstract algebra', there is a question in sec56, exercise3.

It says "The Galois group of a finite extension of a finite field is solvable." is true...

I can't figure out why it's true.

I think this means "every finite polynomial of finite field F is solvable by radicals." (it's correct?)

Help...

 

[mrbohn1]

A finite Galois extension of a finite field has cyclic Galois group, and any cyclic group is solvable.

You are correct in thinking that the term is connected to the solvability of polynomials by radicals: a polynomial is solvable by radicals if and only if it has a solvable galois group.

 

[emptyboat]

Thanks a lot, mrbohn1.

I understand it.

I think 'Cyclic' is deduced by 'primitive element Theorem'.