# Irreducible????

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• May 7th 2006, 09:00 AM
mathlg
Irreducible????
Question:
Determine if x^4 + x^2 + 1 is ireducible in Z3[x]. Factorize it if you can

I think x^4 + x^2 + 1 can be factored as
(x^2 +2)(x^2 +2)

which would make it Irreducible but not sure?

Please Help
• May 7th 2006, 09:03 AM
ThePerfectHacker
Quote:

Originally Posted by mathlg
Question:
Determine if x^4 + x^2 + 1 is ireducible in Z3[x]. Factorize it if you can

I think x^4 + x^2 + 1 can be factored as
(x^2 +2)(x^2 +2)

which would make it Irreducible but not sure?

Please Help

Notice that,
$(x^2+2)(x^2+2)=x^4+4x^2+4$
Since we are in $\mathbb{Z}_3[x]$ we have that,
$x^4+x^2+1$.
Which means it is reducible.

Irreducible over a Field means a polynomial CANNOT be factored into non-constant polynomials as elements of that field. Which is not true here.
• May 7th 2006, 09:12 AM
mathlg
OK I see what you are saying. I forgot about the Z3 [x] which makes it reducible