• Dec 8th 2008, 06:59 PM
universalsandbox
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 PM
ThePerfectHacker
Quote:

Originally Posted by universalsandbox
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].

Write out a complete list of polynomials that are quadradic in \$\displaystyle \mathbb{Z}_3\$.
Then see which of them have zeros.
The ones without zeros are irreducible.
• Dec 9th 2008, 01:22 PM
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.
• Dec 9th 2008, 03:01 PM
ThePerfectHacker
Quote:

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.

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.
• Dec 9th 2008, 05:31 PM
universalsandbox
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 PM
ThePerfectHacker