I would like to know which fields $k$ satisfy the following property:
If $f$ is a a degree $n$ polynomial over $k$ having $n$ distinct roots in $k$, then its derivative $f'$ has $n-1$ distinct roots in $k$.
This problem is inspired by the observation that if $f\in\mathbb{R}[x]$ is a degree $n$ polynomial with $n$ distinct roots, then Rolle's theorem implies that the derivative $f'$ has $n-1$ distinct roots. These $n-1$ roots of $f'$ will lie in the "gaps" between the $n$ roots of $f$. So in particular, $\mathbb{R}$ is an example of a field that satisfies the conditions of the problem.
So far I have mostly been thinking about the case where $k$ is a subfield of $\mathbb{C}$, but I am interested in the general case also.
Here are some observations:
- The field $k$ must have characteristic 0.
This is because in positive characteristic $p$ the polynomial $x^p-x$ has $p\geq2$ distinct roots, but its derivative $-1$ has none.
- The field $k$ cannot contain a root of $x^2 + 1$.
This is because $x^4-1=(x-1)(x+1)(x^2+1)$ would then have four distinct roots, but its derivative $4x^3$ only has one root. So this rules out $\mathbb{C}$, for example.
Another field that works is the intersection $\overline{\mathbb{Q}}\cap\mathbb{R}$ of the algebraic closure of $\mathbb{Q}$ with $\mathbb{R}$. The same Rolle's theorem argument applies here. In fact if $K$ is any algebraically closed subfield of $\mathbb{C}$, then $K\cap\mathbb{R}$ works too.