Past exam of the options I course of the University of Oxford / 2007 / B / B1

The degree $[K:F]$ is the dimension of $K$ as a vector space over $F$. The tower theorem states that for intermediate fields $F\subseteq E\subseteq K$ in a finite extension,If the degree is prime, one factor must be one, forcing $E=K$ or $E=F$; no strictly intermediate field exists.

Let $K$ have 32 elements. Its prime subfield is ${\mathbb{F}}_2$ and $[K:{\mathbb{F}}_2]=5$. Every element outside that subfield has minimal polynomial of degree five, by the tower theorem. Also every member of $K$ satisfies $x^{32}=x$: nonzero elements form a group of order 31. Hence $x^{32}-x$ splits over $K$, with distinct roots because its derivative is $-1=1$ in characteristic two.

Every irreducible degree-five polynomial over ${\mathbb{F}}_2$ divides $x^{32}-x$. Indeed its quotient field has 32 elements, and its residue class of $x$ satisfies that equation. Thus every such polynomial splits into five distinct roots in $K$. Conversely each of the 30 elements outside ${\mathbb{F}}_2$ has one such minimal polynomial, and different minimal polynomials have disjoint root sets. Consequently the number isThere is no extra leading-coefficient factor, since every nonzero leading coefficient over ${\mathbb{F}}_2$ is one.