Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solvable group

  1. May 12, 2010 #1

    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?)

  2. jcsd
  3. May 18, 2010 #2
    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.
    Last edited: May 18, 2010
  4. May 19, 2010 #3
    Thanks a lot, mrbohn1.

    I understand it.

    I think 'Cyclic' is deduced by 'primitive element Theorem'.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Solvable group Date
A On the equivalent definitions for solvable groups Oct 25, 2017
Solvable group: decomposable in prime order groups? Sep 17, 2013
Finite solvable groups Jan 26, 2012
Solvable Groups Mar 29, 2011
Solvable group of matrices? Aug 19, 2008