What would be all the quadratic polynomials that are irreducible in [x]. Integers mod 3. How would you go about finding them. Similarly, how would you find them for [x].
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 is and how to identify them. Thanks.
If irreducible, then T.
If NOT irreducible, then F.
Code:F: x^2 T: x^2+1 F: x^2+2 F: x^2+x F: x^2+x+1 T: x^2+x+2 F: x^2+2x F: x^2+2x+1 T: x^2+2x+2 F: 2x^2 F: 2x^2+1 T: 2x^2+2 F: 2x^2+x T: 2x^2+x+1 F: 2x^2+x+2 F: 2x^2+2x T: 2x^2+2x+1 F: 2x^2+2x+2