Solvable Group: Need Help Understanding Fraleigh's Exercise

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The discussion focuses on the solvability of the Galois group of a finite extension of a finite field, as stated in Fraleigh's 'A First Course in Abstract Algebra.' It is confirmed that this Galois group is cyclic, which implies it is also solvable. The connection to the solvability of polynomials by radicals is emphasized, noting that a polynomial is solvable by radicals if its Galois group is solvable. The primitive element theorem is mentioned as a means to deduce the cyclic nature of the group. Overall, the participants clarify the relationship between Galois groups and polynomial solvability.
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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...
 
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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.
 
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Thanks a lot, mrbohn1.

I understand it.

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