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

2. Originally Posted by universalsandbox
What would be all the quadratic polynomials that are irreducible in $Z_{3}$[x]. Integers mod 3. How would you go about finding them. Similarly, how would you find them for $Z_{5}$[x].
Write out a complete list of polynomials that are quadradic in $\mathbb{Z}_3$.
Then see which of them have zeros.
The ones without zeros are irreducible.

3. 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 $Z_{3}$ is and how to identify them. Thanks.

4. 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 $Z_{3}$ 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 $\mathbb{Z}_3$.
The ones without zeros will be irreducible.

5. 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

6. 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.

7. yep, revised. Thanks.

8. $X^{2}+2$ is also reducible.