A tripling on the algebraic number field
Let \(\ell\) be a prime number and let n be a power of \(\ell\). Let k be an algebraic number field of finite degree which contains a primitive n-th root of unity \(\zeta_ n\). In the present paper the author defines a tripling symbol \((x,y,z)_ n\in <\zeta_ n>\), which is defined on a subset of \(k^{\times}\times k^{\times}\times k^{\times}\) consisting of the elements called strictly orthogonal, and characterizes the norm group from the bicyclic extension \(k(^ n\sqrt{x},^ n\sqrt{y})\). In the case \(k={\mathbb{Q}}\) and \(n=2\), \textit{Y. Furuta} [Nagoya Math. J. 79, 79- 109 (1980; Zbl 0444.12001), cf. ibid. 93, 61-69 (1984; Zbl 0526.12009)] defined a similar symbol and studied it; though this symbol does not coincide with the above, these are essentially the same if these are defined at the same time. For each prime \({\mathfrak l}\) over (\(\ell)\) in k, fix a local Galois extension \(\Omega^{{\mathfrak l}}/k_{{\mathfrak l}}\) whose Galois group is a free pro-\(\ell\) group, where \(k_{{\mathfrak l}}\) is the completion of k by \({\mathfrak l}\). For \(x,y\in k^{\times}\), x and y are said to be orthogonal if its Hilbert symbol \((\frac{x,y}{{\mathfrak p}})_ n=1\) for every \({\mathfrak p}\). In the case \(\ell >2\), x and y are said to be strictly orthogonal if x and y are orthogonal and satisfy that \(x\in N_{k_{{\mathfrak l}}(^ n\sqrt{y})/k_{{\mathfrak l}}}(k_{{\mathfrak l}}(^ n\sqrt{y})^{\times}\cap (\) \(\Omega^{{\mathfrak l}^{\times}})^ n)\) and \(y\in N_{k_{{\mathfrak l}}(^ n\sqrt{x})/k_{{\mathfrak l}}}(k_{{\mathfrak l}}(^ n\sqrt{x})^{\times}\cap (\) \(\Omega^{{\mathfrak l}^{\times}})^ n)\). In the case \(\ell =2\), the notion of strictly orthogonal is defined similarly but needs some more conditions. Let x,y be \(\ell\)-independent and let x,y,z be pairwise strictly orthogonal. Let S be a set of primes of k containing the prime divisors of (\(\ell xyz)\infty\). Then there exists an element \(a\in k(^ n\sqrt{y})\) such that \(a^{l-\sigma}\equiv x\) mod (k(\({}^ n\sqrt{y})^{\times})^ n\) with some additional properties related to S, where \(\sigma \in Gal(k(^ n\sqrt{y})/k)\) such that \(\sigma\) : \({}^ n\sqrt{y}\mapsto \zeta_ n^ n\sqrt{y}\). Take an ideal \({\mathfrak A}\) of k prime to S such that (a)\(\equiv {\mathfrak A}\) (mod n-th power, mod S) in \(k(^ n\sqrt{y})\). Then \((x,y,z)_ n\) is defined as \((\frac{z}{{\mathfrak A}})_ n :\) n-th power residue symbol in k; which is well defined and depends only on x,y,z mod \((k^{\times})^{n'}\), where \(n'=2n\) or \(n'=n\) according to \(\ell =2\) or not. The main purpose of this paper is to prove the following properties of the symbol: It is multiplicative, admits the conjugacy, has the transgression relation, and has the reciprocity law \((x,y,z)_ n^{- 1}=(y,x,z)_ n=(z,y,x)_ n\) under some conditions only when \(\ell =3\). Further it has the norm theorem saying that \(z\in N_{L/k}L^{\times}\), \(L=k(^ n\sqrt{x},^ n\sqrt{y})\) if and only if \((x,y,z)_ n=1\), under the assumption that \([L_{{\mathfrak P}}:k_{{\mathfrak p}}]\leq n\) for every \({\mathfrak P}| {\mathfrak p}| (\ell)\). As applications, the author also gives conditions that \(\ell\) divides the class number of \(k(^ n\sqrt{x},^ n\sqrt{y})\) for some x,y in the cases \(k={\mathbb{Q}}(\zeta_ 3)\) with \(\ell =3\) and \(k={\mathbb{Q}}(\zeta_ 5)\) with \(\ell =5\).
scientific article; zbMATH DE number 3948419