Results 1 to 2 of 2

Thread: Splitting Field of a Polynomial over a Finite Field

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Splitting Field of a Polynomial over a Finite Field

    Hello everyone, I was wondering if I could get some help with this.

    Find the splitting field of the polynomial $\displaystyle f = x^3 + 2x +1 \in \mathbb{Z}_3 [x]$

    Well I know that$\displaystyle \mathbb{Z}_3[x] / \langle x^3 + 2x + 1 \rangle$ is a field extension containing $\displaystyle \alpha = x + \langle f \rangle$ which is a root of the polynomial $\displaystyle f$.

    But is this a splitting field ? Can there not be another element $\displaystyle \alpha '$ which hasn't appeared in this field extension? Thanks for any help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by slevvio View Post
    Hello everyone, I was wondering if I could get some help with this.

    Find the splitting field of the polynomial $\displaystyle f = x^3 + 2x +1 \in \mathbb{Z}_3 [x]$

    Well I know that$\displaystyle \mathbb{Z}_3[x] / \langle x^3 + 2x + 1 \rangle$ is a field extension containing $\displaystyle \alpha = x + \langle f \rangle$ which is a root of the polynomial $\displaystyle f$.

    But is this a splitting field ? Can there not be another element $\displaystyle \alpha '$ which hasn't appeared in this field extension? Thanks for any help.
    Dividing $\displaystyle f(x)=x^3+2x+1$ by $\displaystyle w:=x+<f>$ , we get that

    $\displaystyle x^3+2x+1=(x+2w)(x^2+wx+w^2+2)$ , and

    since the field's characteristic is not 2 we know the above quadratic splits on $\displaystyle \mathbb{Z}/3\mathbb{Z}[x]/<f>$

    iff its discriminant is a square. Now just chek the discriminant indeed is square in this field...

    Tonio

    Pd For example, $\displaystyle w+1$ is another root of $\displaystyle f(x)$ ...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. splitting field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 11th 2010, 04:55 PM
  2. Splitting Field
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Aug 26th 2009, 09:14 AM
  3. Splitting field of an irreducible polynomial
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Jun 25th 2009, 12:14 AM
  4. Field of char p>0 & splitting field
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 22nd 2009, 12:20 AM
  5. polynomial in finite field
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 12th 2008, 08:48 AM

Search Tags


/mathhelpforum @mathhelpforum