What are some examples of irreducible polynomials in Z2[x]?

[kathrynag]

(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]

(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.

 

[eumyang]

How about x² + 1 ?

That is not irreducible in Z₂. x² + 1 = (x + 1)².

 

[SammyS]

Please delete this post.