Results 1 to 2 of 2

Math Help - Galois Theory!

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    27

    Smile Galois Theory!

    Set F=F2(t) where t is a variable, let @ be a root of the equation x^2+x+t=0, set B=f(@) and set b=@^(1/2) and E=F(b). then B is separable over F and E is purely inseparable over B. show that there is no intermediate field B' such that E is separable over B' and B' purely inseparable over F.

    Q2: Suppose that F is a field of characteristic p and that @ is a separable element in some extension. Show that then F(@^p)=F(@)

    Any insights would be great!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by dabien View Post
    Set F=F2(t) where t is a variable, let @ be a root of the equation x^2+x+t=0, set B=f(@) (F@?)and set b=@^(1/2) and E=F(b). then B is separable over F and E is purely inseparable over B.
    The deriative of p(x)=x^2+x+t over \mathbb{F}_2(t) is 1, which implies that the deriative has no root at all. Thus p(x)=x^2+x+t is separable. It follows that B=F(\alpha) is separable over F, where \alpha is a root of p(x). Let k(x)=x^2-\alpha \text{ } (=x^2+\alpha) \in B[x]. We see that k(x) is irreducible in B[x]. However, k(x) is factored as k(x)={(x - \sqrt{\alpha})}^2 in E[x]. Thus E is purely inseparable over B.

    show that there is no intermediate field B' such that E is separable over B' and B' purely inseparable over F.
    [EDIT]

    Q2: Suppose that F is a field of characteristic p and that @ is a separable element in some extension. Show that then F(@^p)=F(@)
    Any insights would be great!
    Observe that \mathbb{F}^p=\mathbb{F}, where \mathbb{F} = F(\alpha) and p is a prime number.
    Last edited by aliceinwonderland; November 22nd 2009 at 10:05 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Textbooks on Galois Theory and Algebraic Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 8th 2011, 06:09 PM
  2. Galois Theory
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 13th 2009, 03:30 AM
  3. Galois Theory
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 11th 2009, 10:39 AM
  4. Galois Theory
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 8th 2009, 08:01 AM
  5. Galois theory
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 28th 2008, 12:54 PM

Search Tags


/mathhelpforum @mathhelpforum