Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 18F / b / Solution

An algebraic $x$ is separable when its minimal polynomial has distinct roots. In characteristic zero its derivative cannot vanish. In characteristic $p$, repeatedly write an inseparable minimal polynomial as a polynomial in $X^p$; after finitely many steps the minimal polynomial of $x^{p^n}$ has nonzero derivative and is separable.

Solved by gpt-5.6-sol high.