Do minimum polynomials always have a nonzero discriminant?

Question

Let $f(x)$ be a minimum polynomial with integer coefficients.

Does $f(x)$ Always have its discriminant equal to nonzero ?

If so , why can't $f(x)$ have a repeated root ?

For all clarity Im talking about minimum polynomials for algebraic integers.

Example : $α = √2 + √3$, then the minimal polynomial in $Q[x]$ is $a(x) = x^4 − 10x^2 + 1 = (x − √2 − √3)(x + √2 − √3)(x − √2 + √3)(x + √2 + √3).$

So I am not talking about linear algebra or matrices.

Answer 1

If you talk about minimal polynomials of endomorphisms/matrices, have a look at $$A=\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}.$$

If you talk about minimal polynomials of algebraic elements over a field, lookup separable vs. inseparable extensions. (In this case, your restriction integer coefficients may imply that you work in characteristic $0$, in which case separability follows; to see this consider $\gcd(f,f')$)

Answer 2

The minimal polynomial $P$ of an algebraic number is irreducible over the base field. Since its formal deriviative $P'$ has $\deg P'<\deg P$ one either has $\gcd(P,P')=1$ or $P'=0$. The latter can only happen in finite characterisitic$~p$, and requires $p\mid \deg P$; if it happens, then $P$ is called inseparable. In the former case, and in particular always when the characterstic is$~0$, the discriminant of$~P$ is nonzero.