This might be somewhat vague, but the Cayley-Hamilton theorem is important in Galois theory. Specifically, if $L/K$ is a finite extension of fields, then an element $a \in L$ has both a minimal polynomial and a characteristic polynomial over $K$. The Cayley-Hamilton theorem yields that the former divides the latter. The two polynomials have their different advantages -- e.g., the minimal polynomial is irreducible and thus can be used to study intermediate fields, whereas the characteristic polynomial depends polynomially on $a$, always has degree $\left[L:K\right]$, and has various other nice functoriality-type properties. The divisibility relation between them makes it possible for one of them to help out the other when necessary.

Concrete example (Proposition 8.6 in Patrick Morandi, Field and Galois theory, Springer 1996, quoted with edits):

Let $K / F$ be a field extension, and let $n = \left[K : F\right] < \infty$. Let $a \in K$, and let $p\left(x\right) = x^m + a_{m-1} x^{m-1} + \cdots + a_1 x^1 + a_0 x^0$ be the monic minimal polynomial of $x$ over $F$. Then, $\operatorname{Norm}_{K/F}\left(a\right) = \left(-1\right)^n a_0^{n/m}$ and $\operatorname{Tr}_{K/F}\left(a\right) = -\dfrac{n}{m} a_{m-1}$.