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Hilbert-symbol equivalence of global function fields. (English) Zbl 0987.11072

Let \(\ell \) be a prime number and \(K\) and \(L\) be global fields of characteristic prime to \(\ell \) and containing primitive \(\ell \)th roots of unity. The author proves that if \(\ell >2\) is a prime then any two global function fields (containing primitive \(\ell \) roots of unity) are degree \(\ell \) Hilbert-symbol equivalent. The degree \(\ell \) Hilbert-symbol equivalence between fields \(K\) and \(L\) is said to be tame at a non-archimedean place \(\mathfrak p\) of \(K\) provided the involved bijection between the sets of all primes of \(K\) and \(L\) preserves the \(\mathfrak p\)-orders modulo \(\ell \). Necessary and sufficient conditions for tame Hilbert-symbol equivalence of global function fields for all prime numbers \(\ell \geq 2\) are also proved. For instance, if \(\ell \) is odd, then \(K\) and \(L\) are degree \(\ell \) tamely Hilbert-symbol equivalent if and only if the zero-degree divisor class groups of \(K\) and \(L\) have the same \(\ell \)-rank.

MSC:

11R58 Arithmetic theory of algebraic function fields
11E12 Quadratic forms over global rings and fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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References:

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