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Math Help - Irreducible Polynomial

  1. #1
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    Irreducible Polynomial

    Prove that the polynomial f(x)=x^2+x+1 is irreducible over <br />
\mathbb{Z}_5<br />
Hence construct
    a field F with 25 elements which extends  \mathbb{Z}_5<br />
. Does the polynomial 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
    x^2+2 = 0
    in F.

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

    g(x) = x^2+2 has no zero in F because this polynomial is also Irreducible over <br />
\mathbb{Z}_5<br />
How do i find solution to 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 f(x)=x^2+x+1 is irreducible over <br />
\mathbb{Z}_5<br />
Hence construct
    a field F with 25 elements which extends  \mathbb{Z}_5<br />
.
    Show the polynomial has no zero, and that will prove it is irreducible, why? This means \mathbb{Z}_5/\left< x^2+x+1 \right> is a field with 5^2 elements. To create a mapping containing this field as a subfield let a\in \mathbb{Z}_5 be mapped to a+\left< x^2+x+1\right>.
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