Brauer groups and character groups of function fields. II
[For part I see ibid. 61, 249-255 (1979; Zbl 0426.13002).] Let F be a global field, E a finitely generated extension field of F, \(K_ n\) a purely transcendental extension of E of transcendence degree n. The authors prove that the Brauer group of \(K_ n\), \(B(K_ n)\), is independent of n: \(B(K_ n)\cong B(K_ m)\) for \(1\leq n,\quad m<\infty.\) The approach is to use the Auslander-Brumer-Fadeev theorem [cf. the authors, Lect. Notes Math. 844, 46-74 (1981; Zbl 0455.12011), p. 51] to write, for any prime p, \(B(K_ n)_ p=B(E)_ p+H_ n,\) ( )\(_ p=p\)- torsion subgroup. Here \(H_ n\) is a direct sum of character groups X(N) of fields N ranging over the finite separable extensions of \(K_ i\); \(0\leq i<n\), where X(N) is the group of continuous homomorphisms from the Galois group of the maximal Abelian extension of N to the discrete group \({\mathbb{Q}}/{\mathbb{Z}}\). A direct argument shows that the divisible part of \(B(K_ n)\) is isomorphic to \(\omega\) copies of \({\mathbb{Q}}/{\mathbb{Z}}\), independent of n. So by standard theory [see \textit{I. Kaplansky}, Infinite abelian groups (1954; Zbl 0057.019)], what must be proved, and is proved, is that the Ulm invariants of \(H_ n\) and \(H_ m\) are equal. - The main work in the paper, then is to study the Ulm invariants of X(K) where K is a field, finitely generated over a global field. The Ulm invariants \(U_ p(\lambda,X(K))\) of X(K) relate to embedding questions: for \(\lambda<\omega\) (\(\omega\) the first infinite ordinal), \(U_ p(\lambda,X(K))\neq 0\) if there is a cyclic p-extension L of K of degree \(p^{\lambda +1}\) which is not embeddable in a cyclic extension of degree \(p^{\lambda +2}\). - A typical result along these lines is that for large finite \(\lambda\), and for K a global field of characteristic not p, \(U_ p(\lambda,X(K))=\omega.\) The last section of the paper is devoted to analogous results for local fields.
scientific article; zbMATH DE number 3865440
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