Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 18F (Galois Theory)

Finite means $[L:K]<\infty$; algebraic means every element satisfies a nonzero polynomial over $K$. Linear dependence among $1,x,\dots ,x^n$ proves finite implies algebraic. The algebraic closure $\overline{\mathbb{Q}}/\mathbb{Q}$ is algebraic but infinite.

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.

Let $\alpha =2^{1/6}=\sqrt{2}/\sqrt[3]{2}$. Then $L=\mathbb{Q}(\alpha ,{\zeta }_3)$ is the splitting field of $X^6-2$. Eisenstein gives $[\mathbb{Q}(\alpha ):\mathbb{Q}]=6$, and adjoining nonreal ${\zeta }_3$ doubles it, so the degree is 12 and the extension is Galois. Valuation at a prime above 2 forces $n\mid 6$ if $x^n=2$; conversely $x={\alpha }^{6/n}$ works for $n=1,2,3,6$.