# Irreducible Polynomial

• Mar 2nd 2008, 07:11 AM
charikaar
Irreducible Polynomial
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?

• Mar 2nd 2008, 07:43 AM
ThePerfectHacker
Quote:

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 $a+\left< x^2+x+1\right>$.