Thread: elementary # theory

Determine if f(x) is irreducible over Z/pZ. If not, factor it.
a) f(x) = x^2 + x + 1 p = 2, 3


b) f(x) = x^3 + 2 p = 3, 5.

Could someone tell me wat exactly does Z/pZ mean???????and how to do this problem?

pZ is the prime number given multiplied by the integers, so the multples of the prime number.

Z/pZ is the ring of integers modulo pZ.

For example, let p=2, then Z/2Z= Z_2= {0,1}. Essentially, you are looking at the remainders when they are divided by 2.

Originally Posted by terr13

pZ is the prime number given multiplied by the integers, so the multples of the prime number.

Z/pZ is the ring of integers modulo pZ.

For example, let p=2, then Z/2Z= Z_2= {0,1}. Essentially, you are looking at the remainders when they are divided by 2.

Hey could you please show me how to do the first question...I am still having troubles doing it. I mean there s nothing alike i could find in the book!

Thanks so much

Originally Posted by felixmcgrady

Determine if f(x) is irreducible over Z/pZ. If not, factor it.
a) f(x) = x^2 + x + 1 p = 2, 3

To see if you can factor this check which elements in Z/pZ are zeros. If p=2 then there are just two elements in Z/pZ and those are the classes [0] and [1]. Note that [0] is not a zero since [0]^2+[0]+[1] = [1]. And [1] is not a zero since [1]^2 + [1]+[1] = [3] = [1]. Therefore it has no zeros. Since it is a quadradic the condition of not having zeros is equivalent to being irreducible. Therefore f(x) is irreducible over Z/pZ for p=2. Now try doing this problem for p=3.