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.
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.