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].

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- Dec 8th 2008, 07:59 PMuniversalsandboxquadratics and polynomials
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].

- Dec 8th 2008, 09:25 PMThePerfectHacker
- Dec 9th 2008, 02:22 PMuniversalsandbox
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. - Dec 9th 2008, 04:01 PMThePerfectHacker
- Dec 9th 2008, 06:31 PMuniversalsandbox
If irreducible, then T.

If NOT irreducible, then F.

Quote:

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

- Dec 9th 2008, 08:42 PMThePerfectHacker
You made three mistakes: x^2+x+1,2x^2+1,2x^2+x+2.

You labeled them as irreducible, while they happen to be reducible. - Dec 9th 2008, 10:39 PMuniversalsandbox
yep, revised. Thanks.

- Dec 10th 2008, 03:11 AMclic-clac
is also reducible.