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Thread: find the field

  1. #1
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    find the field

    Find the field F and the polynomial $\displaystyle f(x)=x^2+ax+b$ from F[x] with the derivation identically equal to 0, f(x) is not the second power of another polynomial.

    Can anybody give me a hint please?
    Thanks
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  2. #2
    MHF Contributor

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    Quote Originally Posted by sidi View Post
    Find the field F and the polynomial $\displaystyle f(x)=x^2+ax+b$ from F[x] with the derivation identically equal to 0, f(x) is not the second power of another polynomial.

    Can anybody give me a hint please?
    Thanks
    such a field has to be an infinite field of characteristic $\displaystyle 2$ and we must also have $\displaystyle a=0.$ here's an example that satisfies the conditions in your problem:

    Spoiler:
    $\displaystyle F=\mathbb{F}_2(t),$ the field of rational functions over $\displaystyle \mathbb{F}_2,$ and $\displaystyle f(x)=x^2+t.$
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