Thread: Irreducible polynomial in field

I have an exam on Monday, and I am not sure about the following.

In the field F= Z5[x] / <x^3 - x^2 - 1> is x^3 - x^2 - 1 irreducible? If not list the irreducible factors.

I am thinking that in F every element is written as something + <x^3 - x^2 - 1> and therefore the polynomial cannot be irreducible because any factor must at least include a multiple of <x^3 - x^2 - 1> and is thus of the same degree.

But my gut tells me that the polynomial should be irreducible....

any help please?

Originally Posted by idontknowlol

In the field F= Z5[x] / <x^3 - x^2 - 1> is x^3 - x^2 - 1 irreducible? If not list the irreducible factors.

Let p = x^3 - x^2 - 1.
Taking on faith that F is actually a field (i.e that p is irreducible in Z5[x]), your question amounts to: Is the zero element/polynomial irreducible?
That's because in Z5[x]/(p), p = 0, or more accurately, p+(p) = 0+(p).
This is one of those special case situations where you have to read the definition of irreducible carefully.

I don't have a text nearby, so I'll use wikipedia:
1) "For any field F, the ring of polynomials with coefficients in F is denoted by F[x]. A polynomial p(x) in is called irreducible over F if it is non-constant and cannot be represented as the product of two or more non-constant polynomials from F[x]."
2) "In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units."

In both cases, the special case of a 0 element would be excluded. Thus I read that as 0 should not called irreducible.

Great. Thanks a lot!