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The Stacks project

Lemma 10.35.12. Let k be a field. Let S be a k-algebra. For any field extension K/k whose cardinality is larger than the cardinality of S we have

  1. for every maximal ideal \mathfrak m of S_ K the field \kappa (\mathfrak m) is algebraic over K, and

  2. S_ K is a Jacobson ring.

Proof. Choose k \subset K such that the cardinality of K is greater than the cardinality of S. Since the elements of S generate the K-algebra S_ K we see that Theorem 10.35.11 applies. \square

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