Results 1 to 3 of 3

Math Help - Calculating the Galois group of a quartic

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    29

    Calculating the Galois group of a quartic

    I've been asked to work out the Galois group of t^4-2t^2-3 \in \mathbb{Q}[x] which I think I can do but then it says "You need not show that any homomorphisms you use are homomorphisms, but should explain why they are members of the Galois group. Write down the subgroups of the Galois group and the intermediate fields of the relevant field extension, and explain why your lists are complete." which has thrown me a bit, not really sure what I'm meant to put?

    To work out the Galois group of t^4-2t^2-3 \in \mathbb{Q}[x] I have said that as t^4-2t^2-3 = (t^2+1)(t^2-1) which are two irreducible quadratics, then the splitting field f(x) is the compositum of \mathbb{Q}(i) and \mathbb{Q}(\sqrt{3}). This is a biquadratic extension and thus the Galois group is V_{4}. Is this right and any help deciphering what else I'm meant to do would be appreciated.

    Is it reffering to the fact that the Klein group V_4 is the product of \mathbb{Z}_2x \mathbb{Z}_2 which are the galois groups of the two quadratics I have. How do I show they are complete?

    Thanks for any advice you can offer!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,577
    Thanks
    1418

    Re: Calculating the Galois group of a quartic

    The problem said "Write down the subgroups of the Galois group". What are the subgroups of V4? What are the corresponding "intemediate fields"?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2010
    Posts
    29

    Re: Calculating the Galois group of a quartic

    Say there are 4 permutations I, P_1, P_2, P_3 with P_3=P_1P_2 then there are five subgroups:

    \{I\}, corresponding to \mathbb{Q}(i, \sqrt{3})

    \{I,P_1\}, corresponding to \mathbb{Q}(\sqrt{3}) As this fixes \sqrt{3} say

    \{I,P_2\}, corresponding to \mathbb{Q}(i) As this fixes i say

    \{I,P_3\}, corresponding to \mathbb{Q}(i\sqrt{3}) As this fixes i\sqrt{3}

    and \{I,P_1,P_2,P_3\} corresponding to \mathbb{Q}

    I think thats right??
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Calculating Galois Groups....
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 18th 2010, 09:19 AM
  2. Galois Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 23rd 2009, 06:08 PM
  3. Galois Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 17th 2009, 01:33 PM
  4. Galois group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 28th 2009, 12:40 AM
  5. Galois group of f(x)=x^4-5x^2+6
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 3rd 2008, 08:37 PM

Search Tags


/mathhelpforum @mathhelpforum