Math Help - Irreducibility over Integers mod p

1. Irreducibility over Integers mod p

Prove that x^3-9 is irreducible over the integers mod 31.

Prove that x^3-9 is reducible over the integers mod 11.

Any help in this would be greatly appreciated as I study for my final exam.

2. Originally Posted by apalmer3
Prove that x^3-9 is irreducible over the integers mod 31

Prove that x^3-9 is reducible over the integers mod 11.

Any help in this would be greatly appreciated as I study for my final exam.
It is sufficient here to prove the polynomials have no zeros.

3. Okay! I see how that helps prove that it's irreducible over the integers mod 31... but how does that prove that it's reducible over integers mod 11?

Thanks for the help!

4. Actually... I don't understand...

For example:
x^2+1 is reducible mod 5. It's equal to (x+2)(x+3). How can that be when (real) roots don't exist?

5. Originally Posted by apalmer3
but how does that prove that it's reducible over integers mod 11?
Note $4$ is a zero, thus, $(x-4)$ is a factor of $x^3 - 9$.

6. Originally Posted by apalmer3
Actually... I don't understand...

For example:
x^2+1 is reducible mod 5. It's equal to (x+2)(x+3). How can that be when (real) roots don't exist?
That only works for polynomials over $\mathbb{R}$. You are in a different field, i.e. mod 5.

7. Originally Posted by ThePerfectHacker
That only works for polynomials over $\mathbb{R}$. You are in a different field, i.e. mod 5.

Okay. But (x+2)(x+3) = x^2+5x+6

In the integers mod 5, that equals x^2+0x+1 = x^2+1... see what I mean?