Results 1 to 2 of 2

Math Help - Galois Group

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    62

    Galois Group

    Find the Galois group of  f(x)=(x^3-2)(x^3-3) .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by mathman88 View Post
    Find the Galois group of  f(x)=(x^3-2)(x^3-3) .
    I could have this wrong, but here goes:

    The splitting field for  f(x) is  K=\mathbb{Q}(\sqrt[3]2,\sqrt[3]3,\rho) , where  \rho is a third root of unity.

    We then have  [K:\mathbb{Q}]=18

    Since we can determine all automorphisms based on the roots of  f(x) and  \alpha\mapsto\alpha' , where  \alpha' is another root of  m_{\alpha,\mathbb{Q}}(x) , let's see where that takes us.
    Define the automorphisms as follows:

     \sigma:\begin{cases} \sqrt[3]2\mapsto\rho\sqrt[3]2\\ \rho\mapsto\rho\\ \sqrt[3]3\mapsto\sqrt[3]3 \end{cases} \quad\quad \tau:\begin{cases} \sqrt[3]2\mapsto\sqrt[3]2\\ \rho\mapsto\rho^2\\ \sqrt[3]3\mapsto\sqrt[3]3 \end{cases} \quad\quad \gamma:\begin{cases} \sqrt[3]2\mapsto\sqrt[3]2\\ \rho\mapsto\rho\\ \sqrt[3]3\mapsto\rho\sqrt[3]3 \end{cases}

    Now note  |\sigma|=3,\; |\tau|=2,\; |\gamma|=3 , so  |<\sigma,\tau,\gamma>|=18= [K:\mathbb{Q}] = |\text{Gal}(K/\mathbb{Q})| .

    I'll leave it to you to see why  <\sigma,\tau> \cong S_3 .

    So I'm thinking  \text{Gal}(K/\mathbb{Q}) = <S_3,\gamma> , where  \gamma is a three-cycle.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Galois group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 25th 2010, 10:39 PM
  2. galois group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 19th 2010, 04:10 AM
  3. galois group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 14th 2010, 12:26 AM
  4. Galois group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 20th 2010, 03:07 AM
  5. Galois Group, FTG
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 15th 2009, 09:25 PM

Search Tags


/mathhelpforum @mathhelpforum