# Irreducible Polynomial

• Mar 2nd 2008, 07:11 AM
charikaar
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.
• Mar 2nd 2008, 07:43 AM
ThePerfectHacker
Quote:

Originally Posted by charikaar
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 $\displaystyle a+\left< x^2+x+1\right>$.