What would be all the quadratic polynomials that are irreducible in $\displaystyle Z_{3}$[x]. Integers mod 3. How would you go about finding them. Similarly, how would you find them for $\displaystyle Z_{5}$[x].

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- Dec 8th 2008, 06:59 PMuniversalsandboxquadratics and polynomials
What would be all the quadratic polynomials that are irreducible in $\displaystyle Z_{3}$[x]. Integers mod 3. How would you go about finding them. Similarly, how would you find them for $\displaystyle Z_{5}$[x].

- Dec 8th 2008, 08:25 PMThePerfectHacker
- Dec 9th 2008, 01:22 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 $\displaystyle Z_{3}$ is and how to identify them. Thanks. - Dec 9th 2008, 03:01 PMThePerfectHacker
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

The ones without zeros will be irreducible. - Dec 9th 2008, 05:31 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, 07:42 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, 09:39 PMuniversalsandbox
yep, revised. Thanks.

- Dec 10th 2008, 02:11 AMclic-clac
$\displaystyle X^{2}+2$ is also reducible.