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**Higher degree Hilbert-symbol equivalence of number fields.**
*(English)*
Zbl 0978.11058

Quadratic Hilbert-symbol equivalence has been studied so far as a convenient set of local conditions for the Witt equivalence (i.e. the existence of an isomorphism between Witt rings of quadratic forms) of global fields. The authors generalize the set-up to higher degree Hilbert symbols and study the higher degree Hilbert-symbol equivalence. In the absence of Witt ring theory for higher dimensional forms the authors use a substitute for Witt equivalence called the Harrison equivalence, and they compare the higher degree Hilbert-symbol equivalence and higher degree Harrison equivalence of number fields.

More specifically, let \(n \geq 2\) be a fixed integer and assume throughout that all fields considered contain primitive \(n\)th roots of unity.

Two number fields \(K\) and \(L\) are said to be degree \(n\) Hilbert-symbol equivalent when there is an isomorphism \(f:\mu _n(K) \to \mu _n(L)\) of the groups of \(n\)th roots of unity, a group isomorphism \(t:K^*/K^{*n} \to L^*/L^{*n}\) and a bijective map \(T:\Omega (K) \to \Omega (L)\) between the sets of all primes of the two fields, preserving Hilbert symbols of degree \(n\), \[ (a,b)^f_P = (ta,tb)_{TP} \] for all \(n\)-th power classes \(a,b \in K^*/K^{*n}\) and all primes \(P \in \Omega (K).\)

A Harrison map of degree \(n\) is a group isomorphism \(t:K^*/K^{*n} \to L^*/L^{*n}\) sending the coset of \(-1\) onto the coset of \(-1\), and preserving the norm groups of cyclic extensions of degree \(n.\) If such a map exists, the fields \(K\) and \(L\) are said to be degree \(n\) Harrison equivalent.

The main result of the paper states that, for any \(n\), Hilbert-symbol equivalence of degree \(n\) implies the Harrison equivalence of degree \(n\), and when \(n\) is a prime, the two equivalences coincide.

The authors’ approach is based on a proof of these results for \(n=2\) [the reviewer, Tatra Mt. Math. Publ. 11, 7-16 (1997; Zbl 0978.11012)].

More specifically, let \(n \geq 2\) be a fixed integer and assume throughout that all fields considered contain primitive \(n\)th roots of unity.

Two number fields \(K\) and \(L\) are said to be degree \(n\) Hilbert-symbol equivalent when there is an isomorphism \(f:\mu _n(K) \to \mu _n(L)\) of the groups of \(n\)th roots of unity, a group isomorphism \(t:K^*/K^{*n} \to L^*/L^{*n}\) and a bijective map \(T:\Omega (K) \to \Omega (L)\) between the sets of all primes of the two fields, preserving Hilbert symbols of degree \(n\), \[ (a,b)^f_P = (ta,tb)_{TP} \] for all \(n\)-th power classes \(a,b \in K^*/K^{*n}\) and all primes \(P \in \Omega (K).\)

A Harrison map of degree \(n\) is a group isomorphism \(t:K^*/K^{*n} \to L^*/L^{*n}\) sending the coset of \(-1\) onto the coset of \(-1\), and preserving the norm groups of cyclic extensions of degree \(n.\) If such a map exists, the fields \(K\) and \(L\) are said to be degree \(n\) Harrison equivalent.

The main result of the paper states that, for any \(n\), Hilbert-symbol equivalence of degree \(n\) implies the Harrison equivalence of degree \(n\), and when \(n\) is a prime, the two equivalences coincide.

The authors’ approach is based on a proof of these results for \(n=2\) [the reviewer, Tatra Mt. Math. Publ. 11, 7-16 (1997; Zbl 0978.11012)].

Reviewer: Kazimierz Szymiczek (Katowice)

### MSC:

11R21 | Other number fields |

11R20 | Other abelian and metabelian extensions |

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

19F15 | Symbols and arithmetic (\(K\)-theoretic aspects) |