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Thread: Irreducible Polynomial

  1. #1
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    Irreducible Polynomial

    Prove that the polynomial $\displaystyle f(x)=x^2+x+1$ is irreducible over $\displaystyle
    \mathbb{Z}_5
    $ Hence construct
    a field F with 25 elements which extends $\displaystyle \mathbb{Z}_5
    $. Does the polynomial $\displaystyle g(x) = x^2+2$ have
    a zero in F? In other words, you need to find all the solutions (if any) to the equation
    $\displaystyle x^2+2 = 0$
    in F.

    My attempt.
    $\displaystyle f(x)=x^2+x+1$ has no zero thus irreducible over $\displaystyle
    \mathbb{Z}_5
    $ (treid x=0,1,2,3,4) How do i construct field with 25 elements?

    $\displaystyle g(x) = x^2+2$ has no zero in F because this polynomial is also Irreducible over $\displaystyle
    \mathbb{Z}_5
    $ How do i find solution to $\displaystyle x^2+2 = 0$
    in F?

    Thank you in advance.
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Prove that the polynomial $\displaystyle f(x)=x^2+x+1$ is irreducible over $\displaystyle
    \mathbb{Z}_5
    $ Hence construct
    a field F with 25 elements which extends $\displaystyle \mathbb{Z}_5
    $.
    Show the polynomial has no zero, and that will prove it is irreducible, why? This means $\displaystyle \mathbb{Z}_5/\left< x^2+x+1 \right>$ is a field with $\displaystyle 5^2$ elements. To create a mapping containing this field as a subfield let $\displaystyle a\in \mathbb{Z}_5$ be mapped to $\displaystyle a+\left< x^2+x+1\right>$.
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