The Chevalley-Bass Theorem

Abstract

This is an exposition of a theorem due to Chevalley (1951) and Bass (1965). Let $K$ be a finitely generated field. Then there exists a positive integer $\Lambda$, depending only on $K$, such that for every positive integer $n$ the following holds: if $\alpha\in K$ is a $\Lambda n$th power in the cyclotomic extension $K(\zeta_{\Lambda n})$, then $\alpha$ is an $n$th power in $K$. We also give explicit expressions for a suitable $\Lambda$ of two kinds: one in terms of the degree of the maximal abelian subfield of $K$, the other in terms of the discriminant of this subfield.