Hilbert symbol equivalence of global fields. (Q2753378)

In this booklet, the main object of interest is the equivalence of global fields with respect to Hilbert symbol equivalence (HSE for short). Let \(K\) and \(L\) be global fields both containing a primitive \(n\)th root of unity. A degree \(n\) HSE between \(K\) and \(L\) is a triple \((T,t,f)\), where \(T\) is a bijection between sets of places of \(K\) and \(L\), \(t\) is an isomorphism between the \(n\)th power class groups \(K^*/K^{*n}\) and \(L^*/L^{*n}\), and \(f\) is an isomorphism between \(\mu_n(K)\) and \(\mu_n(L)\) such that \(f((a,b)_{n,P})=(ta,tb)_{n,TP}\) for any prime \(P\) of \(K\) and any \(a,b\in K^*.\) The notion of HSE arose in the theory of quadratic forms over global fields. The degree 2 HSE was introduced in [\textit{R. Perlis}, \textit{K. Szymiczek}, \textit{P. E. Conner}, and \textit{R. Litherland}, Contemp. Math. 155, 365-387 (1994; Zbl 0807.11024)] as a necessary and sufficient condition for the Witt equivalence. NEWLINENEWLINENEWLINEThe generalization to degree \(n>2\) first appeared in the joint paper of the author and the reviewer [\textit{A. Czogała} and \textit{A. Sładek}, Tatra Mt. Math. Publ. 11, 77-88 (1997; Zbl 0978.11058)] and turned out to be a necessary and sufficient condition for two global fields to have isomorphic graded Milnor rings modulo \(n\). When \(n\) is prime, HSE can be rephrased in terms of field invariants. For \(n=2\) see [\textit{K. Szymiczek}, Commun. Algebra 19, 1125-1149 (1991; Zbl 0724.11020)], whereas for odd prime \(n\) one can find it in [\textit{A. Czogała} and \textit{A. Sładek}, J. Number Theory 72, 363-376 (1998; Zbl 0922.11096)]. NEWLINENEWLINENEWLINEThe above resulted in the classification of global fields with respect to HSE. The HSE \((T,t,f)\) of degree \(n\) is called tame with respect to a prime \(P\) of \(K\) if \(\text{ord}_P(a)\equiv\text{ord}_{TP}(ta)\pmod n\) for any \(a\in K^*.\) The HSE is called tame if it is tame with respect to any prime \(P.\) The tame HSE of degree 2 was studied for example by the author in [Acta Arith. 58, 27-46 (1991; Zbl 0733.11012)] whereas the higher degree generalization of the tame equivalence was introduced and discussed by the author in [Abh. Math. Semin. Univ. Hamb. 69, 175-185 (1999; Zbl 0968.11038)]. NEWLINENEWLINENEWLINEThe paper is an excellent and self-contained presentation of all known results on HSE. The author concentrates mainly on HSE of degree \(n>2\) whereas the results in case \(n=2\) are just carefully cited. The reader is provided not only with a unified treatment of all details of the theory but also with some new results that has been obtained by improvements of methods presented in earlier papers. For example, to get an HSE between \(K\) and \(L\), one defines it on a subset \(S\) of primes of the field \(K\) and then extends it step by step to oversets of \(S.\) The author succeeds in controlling at each step the number of wild (not tame) primes, and in that way he gets some bounds on the number of all wild primes. Other new results are connected with interpretation of \(S\)-tame equivalence in Milnor K-theory and the interpretation of so called \(S\)-integral equivalence in the theory of quadratic forms.