Factoring polynomials

[arfa brane]

This is a question from a previous exam:

Express the following polynomial in Z₃[x] as a product of irreducibles:

p(x) = 2x⁴ + x² + x + 2.

I can divide this by x + 2, but x + 2 is zero when x = 1 (in Z₃[x]), so it's not irreducible, right?

[mathman]

This is a question from a previous exam:

Express the following polynomial in Z₃[x] as a product of irreducibles:

p(x) = 2x⁴ + x² + x + 2.

I can divide this by x + 2, but x + 2 is zero when x = 1 (in Z₃[x]), so it's not irreducible, right?

In this context what do you mean by irreducible? Ordinarily, a linear factor is irreducible.

[rpenner]

Proof that the last term is not factorisable.
If it was factorisable, it would be into terms of order 1 and 2. So it suffices to check all irreducible monomials polynomials with order 1.

[arfa brane]

Ordinarily, a linear factor is irreducible.

Yeah, trick question.
Irreducible means it can't be written as a product of lower degree polynomials.

Proof that the last term is not factorisable.

The other way to prove is not factorisable is to show it has no zeros in Z₃[x], which clearly it doesn't. This only applies to degree 3 or 2 polynomials (why?).

[rpenner]

A degree four polynomial could be a product of two irreducible polynomials of degree 2.

Examples (in Z_3[x]):
x^4 + 1
x^4 + 2x^2 + 1
x^4 + x^3 + x + 2
x^4 + x^3 + 2 x^2 + 2 x + 1
x^4 + 2 x^3 + 2x + 2
x^4 + 2 x^3 + 2 x^2 + x + 1

So factoring in Z_3[x] an polynomial like x^8 + 2 can be tricky.