= Bounded-degree extensions with restricted ramification
Fix a <number field> $K$, a positive integer $d$, and a finite set $S$ of its finite places. There are finitely many <field extensions> $L/K$ of degree at most $d$ unramified outside $S$, inside a fixed <algebraic closure>. To see this, view them as extensions of the <rational numbers> of bounded degree. Their ramified rational primes belong to a fixed finite set. At each of these primes, the <different exponent> is bounded in terms of the local degree and ramification index, giving a bound for the absolute <field discriminant>. The <Hermite–Minkowski theorem> gives finitely many fields up to isomorphism, and each has only finitely many embeddings into the fixed closure. This useful form of arithmetic compactness is stated and proved for $K=\mathbb Q$ in https://math.mit.edu/classes/18.785/2018fa/LectureNotes14.pdf[Andrew Sutherland's number-theory notes, Theorem 14.25]; the preceding reduction gives the general case.
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