= Solution
The <tower law for field extensions> states that, for $F\subseteq K\subseteq L$ with both successive degrees finite,
$$
\boxed{[L:F]=[L:K][K:F].}
$$
Indeed, if $u_i$ form an $F$-<basis> of $K$ and $v_j$ form a $K$-basis of $L$, their products $u_iv_j$ 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 \b[doubling the cube by straightedge and compass is impossible].
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