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Math Help - Splitting Field of a Polynomial over a Finite Field

  1. #1
    Senior Member slevvio's Avatar
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    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 f = x^3 + 2x +1 \in \mathbb{Z}_3 [x]

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

    But is this a splitting field ? Can there not be another element \alpha ' which hasn't appeared in this field extension? Thanks for any help.
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    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 f = x^3 + 2x +1 \in \mathbb{Z}_3 [x]

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

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

    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 \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, w+1 is another root of f(x) ...
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