Past exam of the options I course of the University of Oxford / 2013 / A (Introduction to Fields)

The degree of a field extension is the dimension of the larger field as a vector space over the smaller one:The field extension is a finite field extension precisely when this dimension is finite, equivalently when finitely many elements of $K$ span $K$ over $F$. Finiteness here concerns the dimension, not the number of elements of the fields; for example, $\mathbb{C}$ has degree two over $\mathbb{R}$ although both fields are infinite.

Let $[L:F]=n<\infty$. The intermediate field $K$ is an $F$-linear linear subspace of $L$, so the finite-dimensional linear subspace theorem gives $[K:F]\le n$. To handle the other field extension, take an $F$-basis $v_1,\dots ,v_n$ of $L$. Every $\ell \in L$ is a sum $\sum f_i{v}_i$ with $f_i\in F\subseteq K$, so the same list spans $L$ as a $K$-vector space. It need not remain linearly independent, but spanning suffices to give $[L:K]\le n$. Hence both intermediate extensions are finite.

The tower law for field extensions states that, for $F\subseteq K\subseteq L$ with both successive degrees finite,Indeed, if $u_i$ form an $F$-basis of $K$ and $v_j$ form a $K$-basis of $L$, their products $u_i{v}_j$ form an $F$-basis of $L$: expansion gives spanning, and independence follows first over $K$ and then over $F$.

Starting from a unit segment, every straightedge and compass construction introduces coordinates by solving linear or quadratic equations over the coordinates already available. Thus a constructible number belongs to a tower of field extensions whose successive degrees are one or two; the total degree is a power of two. Doubling the volume of a unit cube would require the edge length $\sqrt[3]{2}$. Its minimal polynomial over $\mathbb{Q}$ is $x^3-2$, irreducible by Eisenstein's criterion at the prime number two. Thus $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$. If that number lay in a construction field of degree $2^r$, the tower law would make three divide $2^r$, a contradiction. Therefore doubling the cube by straightedge and compass is impossible.