1. ## Polynomial

I have the polynomial $\displaystyle f(x) = x^4-10x^2+1$ over $\displaystyle \mathbb Q$. It can be easily seen that Klein's Four group is the Galois group of this polynomial.
How can I show that the corresponding polynomial
$\displaystyle g(x) \in \mathbb F_p[x]$ is irreducible in $\displaystyle \mathbb F_p[x]$ for every prime $\displaystyle p$.

2. Oh man, please, it's polynomial !

3. This is not true. If you take $\displaystyle p=3$, $\displaystyle x^4 - 10x^2 + 1 = (x^2+1)^2$. I think you want to prove that this polynomial is always reducible as opposed to irreducible. One way to proved is by Dedekind's theorem (again) and by an understanding of transitive subgroups of $\displaystyle S_4$. The transitive subgroup $\displaystyle V$ (Klein four group) consists of an identity element and $\displaystyle 2$-cycle products ($\displaystyle V=\{\text{id},(12)(34),(13)(24),(14)(23)\}$). Therefore, mod any $\displaystyle p$ the polynomial $\displaystyle x^4-10x^2+1$ either splits or factors into quadradic factors. Therefore, it is always reducible.

4. Sorry what I meant was reducible not irreducible, it was a mistake.

5. Originally Posted by ZetaX
I have the polynomial $\displaystyle f(x) = x^4-10x^2+1$ over $\displaystyle \mathbb Q$. It can be easily seen that Klein's Four group is the Galois group of this polynomial.
How can I show that the corresponding polynomial
$\displaystyle g(x) \in \mathbb F_p[x]$ is reducible in $\displaystyle \mathbb F_p[x]$ for every prime $\displaystyle p$.
Here is another way of proving this which is elementary. Let $\displaystyle p$ be prime with $\displaystyle p\geq 7$.

Notice the following factorizations:
$\displaystyle x^4-10x^2+1 = (x^2 + 2\sqrt{2}x-1)(x^2-2\sqrt{2}x-1)$ [1]
$\displaystyle x^4-10x^2+1 = (x^2 + 2\sqrt{3}x+1)(x^2 - 2\sqrt{3}x+1)$ [2]
$\displaystyle x^4-10x^2+1 = (x^2 - 5 + 2\sqrt{6})(x^2 - 5 - 2\sqrt{6})$ [3]

If $\displaystyle 2$ is a square root in $\displaystyle \mathbb{F}_p$ then we can use the factorization in [1] where we replace $\displaystyle \sqrt{2}$ by a square root of $\displaystyle 2$ modulo $\displaystyle p$. If $\displaystyle 3$ is a square root in $\displaystyle \mathbb{F}_p$ then we can use the factorization in [2] where we replace $\displaystyle \sqrt{3}$ by a square root of $\displaystyle 3$ modulo $\displaystyle p$. If neither $\displaystyle 2,3$ are squares mod $\displaystyle p$ then it means $\displaystyle (2/p),(3/p)=-1$ (Legendre symbol). However, this means $\displaystyle (6/p) = (2/p)(3/p) = (-1)(-1) = 1$. Therefore, $\displaystyle 6$ would be a square in $\displaystyle \mathbb{F}_p$ and so we can use the factorization in [3].

Therefore, $\displaystyle \overline{f(x)}\in \mathbb{F}_p[x]$ is always reducible for any $\displaystyle p$.

6. why should p be greater than or equal to 7?

7. Originally Posted by ZetaX
why should p be greater than or equal to 7?
Because I just wanted $\displaystyle (2,p)=(3,p) = 1$ if $\displaystyle p=2,3$ then they are not relatively prime anymore. The cases $\displaystyle p=2,3$ can be checked seperately. It seems I did not need $\displaystyle p\geq 7$ but rather $\displaystyle p\geq 5$.