Irreducible Polynomial

**charikaar** · March 2nd 2008, 07:11 AM

Prove that the polynomial $f(x)=x^2+x+1$ is irreducible over $ \mathbb{Z}_5 $ Hence construct
a field F with 25 elements which extends $\mathbb{Z}_5 $ . 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 $ \mathbb{Z}_5 $ (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 $ \mathbb{Z}_5 $ How do i find solution to $x^2+2 = 0$
in F?

Thank you in advance.

**ThePerfectHacker** · March 2nd 2008, 07:43 AM

Originally Posted by charikaar

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

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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