Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 18H (Galois Theory)

The polynomial $X^m-1$ is separable, since its derivative $m{X}^{m-1}$ has no common root with it when the characteristic does not divide $m$. Its roots form a cyclic multiplicative group of order $m$. For completeness, any finite subgroup of a field's multiplicative group is cyclic: if its exponent is $e$, all its elements are roots of $X^e-1$, so its size is at most $e$, while the elementary decomposition of a finite abelian group gives an element of order $e$.

Choose a primitive root $\zeta$. Then $L=K(\zeta )$, and every field automorphism sends $\zeta$ to ${\zeta }^a$ for a unique unit $a$ modulo $m$. Composition multiplies the exponents, and fixing $\zeta$ fixes all of $L$. This is the required injective homomorphismFor $K={\mathbb{F}}_q$, the least extension ${\mathbb{F}}_{q^d}$ containing a primitive $m$th root is characterized by $m\mid q^d-1$, since its multiplicative group is cyclic of order $q^d-1$. Thus $[L:K]$ is the multiplicative order of $q$ modulo $m$. The successive powers of $4$ modulo $11$ are $4,5,9,3,1$, so