It is difficult to calculate the algebraic K-theory of rings, but the situation simplifies after K(1)-localization (at a given prime p). A result of Thomason identifies the K(1)-local algebraic K-theory of a Z[1/p]-algebra as essentially the topological K-theory of the \'etale homotopy type (e.g., the usual homotopy type for algebras over the complex numbers). We show that for an arbitrary ring R, the K(1)-local K-theory of R is equivalent to that of R[1/p]. For p-adic rings, we expect that arises from a type of comparison between p-adic \'etale and crystalline-type cohomology theories. Our result relies on the cyclotomic trace K \to TC and recent advances of Bhatt-Morrow-Scholze and Bhatt-Scholze around TC. This is joint work with Bhargav Bhatt and Dustin Clausen.