Thread: Irreducible polynomials

Hi, does anyone know how to do the following questions?

In each of the following cases, find all roots of f(x) in F and factor f(x) as the product of irreducible polynomials in F[x].

(a) F=R, f(x) = (x^6)-1 in R[x]
(b) F=Z3, f(x) = (x^5)+x+1 in Z3[x]
(c) F=Z5, f(x) = (x^3)+2x+3 in Z5[x]

Thanks very much guys....

Originally Posted by suedenation

Hi, does anyone know how to do the following questions?

In each of the following cases, find all roots of f(x) in F and factor f(x) as the product of irreducible polynomials in F[x].

(a) F=R, f(x) = (x^6)-1 in R[x]

You have, thus find all real values such as . This only happens for (note the other 4 roots are complex we are only concerned with real). Thus, and are factors. Dividing by we get,
. Thus, the polynomial fractors as
in

Originally Posted by suedenation

(b) F=Z3, f(x) = (x^5)+x+1 in Z3[x]

To factor in we simply see what are its zero's. This is easy to check because they are only 3 elements in this field. Checking we see that 1 is a zero because, under addition and multiplication modulo 3. Thus, is a factor. Dividing these polynomials we get, (working modulo 3).

Checking the zero's of this polynomial we find that 1 is it zero. Thus, is a factor. Dividing these polynomials we get, (working modulo 3).
, notice this polynomial has no zero and because it has a degree of 3 it is irreducible (this is a theorem about the irreducibility of 2 and 3 degree polynomials). Thus, it factors as,
or another way of writing this,

I do part (c) latter unless you understand and I could stop.