Proving that the polynomial $f(x)$ is separable

Question

Let $f(x)=x^{4}-10x^{2}-25\in \mathbb Q[x]$. I want to prove that $f(x)$ is a separable polynomial over $\mathbb Q$. I know that from the definition $f(x)$ is separable if none of the irreducible factors of $f(x)$ in $F[x]$ has a repeated root in a splitting field for $f(x)$ over $F$.

So for my example here $f(x)=x^{4}-10x^{2}-25$ $F=\mathbb Q$ and $F[x]=\mathbb Q[x]$

so do i just need to factorise $f(x)$ and show that the factors of $f(x)$ has no repeated roots in $\mathbb Q$?

Answer 1

No. You only have to prove $f(x)$ and its derivative have no common root in $\mathbf C$. So you just have to compute $\;\gcd(f, f')$ by Euclid's algorithm and check it is $1$.

Here, taking into account the particular value of $f(x)$, and $f'(x)=4x^3-20x=4x(x^2-5$, you only have to show that none of the irreducible factors of $f'(x)$ ($x$ and $x^2-5$ – divides $f(x)$). It amounts to checking neither $0$ nor $\sqrt 5$ is a root of $f$.

Answer 2

Note that $f(x)=x^4-10x^2-25=(x^2-5)^2-50$ has four distinct zeros given by $$\pm\sqrt{5\pm\sqrt{50}}.$$