Originally Posted by
universalsandbox I'm confused on just how to write them out.
I know that integers mod p, for p prime, gives a field.
and for F, finite, |F| = p^n.
In this case, n = quadratic, p=3
|F| = 3^2 = 9
and there are p(p+1)/2 = 3(3+1)/2 = 6 reducible ones.
So there should be 9-6 = 3 irreducible ones. But could someone give me the entire list to get an idea of what a quadratic under $\displaystyle Z_{3}$ is and how to identify them. Thanks.
The quadradics are:
Code:
x^2
x^2+1
x^2+2
x^2+x
x^2+x+1
x^2+x+2
x^2+2x
x^2+2x+1
x^2+2x+2
2x^2
2x^2+1
2x^2+2
2x^2+x
2x^2+x+1
2x^2+x+2
2x^2+2x
2x^2+2x+1
2x^2+2x+2
Now check which ones has zeros in $\displaystyle \mathbb{Z}_3$.
The ones without zeros will be irreducible.