# Irreducible Polynomials

(a) Find all irreducible polynomials of degree less than or equal to 3 in Z2[x].
(b) Show that f(x) = x4 + x + 1 is irreducible over Z2.
(c) Factor g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x].

We have an irreducible polynomial if it cannot be factored into a product of polynomials of lower degree.
a)deg 1: x, x+1
deg 2: x^2+x+1
deg 3: x^3+x^2 + 1, x^3 + x +1
b) and c) get me confused.
I know a polynomial in F[x] is irreduble over F iff for all f(x),g(x) in F[x], p(x)|f(x)g(x) implies p(x)|f(x) or p(x)|g(x).
I don't know if that helps or if there is a simpler way to do this with degrees or something

SammyS
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(a) Find all irreducible polynomials of degree less than or equal to 3 in Z2[x].
(b) Show that f(x) = x4 + x + 1 is irreducible over Z2.
(c) Factor g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x].

We have an irreducible polynomial if it cannot be factored into a product of polynomials of lower degree.
a)deg 1: x, x+1
deg 2: x^2+x+1
[STRIKE]How about x2 + 1 ?[/STRIKE] (See eumyang's post.)
deg 3: x^3+x^2 + 1, x^3 + x +1
b) and c) get me confused.
For (b): You have all of the irreducible polynomials of degree less than or equal to 3 in Z2[x]. Show that none of them divides f(x) = x4 + x + 1 .

For (c): Find one of the polynomials of degree less that 4, which divides g(x) = x5 + x + 1 .
I know a polynomial in F[x] is irreducible over F iff for all f(x),g(x) in F[x], p(x)|f(x)g(x) implies p(x)|f(x) or p(x)|g(x).
I don't know if that helps or if there is a simpler way to do this with degrees or something
I'm assuming the Z2 means Z2, the integers mod 2.

Last edited:
eumyang
Homework Helper
How about x2 + 1 ?
That is not irreducible in Z2. x2 + 1 = (x + 1)2.

SammyS
Staff Emeritus