Sur quelques questions dans la théorie des corps biquadratiques
Let \(K(\alpha)\) denote an algebraic field which is biquadratic over the arbitrary ground field \(K\), and let \(C\) denote a cyclic field, \(G\) a non-cyclic Galois field and \(N\) a non-Galois field. Let further \(\rho\) denote the number of real conjugates of \(\alpha\), \(m\) the total number of quadratic subfields over \(K\) of \(K(\alpha)\) and \(\mu\) the number of real ones among these. In an earlier paper [Avh. Norske Vid. Akad. Oslo 1937, No. 4, 1--26 (1937; Zbl 0017.10003)] the author derived some theorems concerning the possible quadratic subfields of \(K(\alpha)\), and these investigations are given a second time. Next he obtains a classification of all \(K(\alpha)\) by means of \(m\), \(C\), \(G\) and \(N\), and further a more detailed division if \(K\) is real, based on \(m\), \(C\), \(G\), \(N\), \(\rho\) and \(\mu\). In the last case there are 11 various types of fields \(K(\alpha)\). It is proved that there exists an infinity of fields belonging to all these classes. In a second part of the paper the author shows how to determine a system of fundamental units in the classes \(5-11\), where \(\mu\geq 1\), by means of the knowledge of the fundamental units in the real quadratic subfields. For the cyclic fields (classes 6 and 9) a detailed investigation is to be found in a paper of \textit{H. Hasse} [Abh. Deutsch. Akad. Wiss. Berlin, Math.-Naturw. Kl. 1948, No. 2, 95 p. (1950; Zbl 0035.30502)], and for the classes 10 and 11 in a paper of \textit{T. Kubota} [Nagoya Math. J. 10, 65--85 (1956; Zbl 0074.03001)]. As an example of the results obtained we quote the following theorem concerning the group of units for class 7, where \((m, \rho, \mu)=(1, 2, 1)\) and \(K(\alpha)\) is of type \(N\): A pair of fundamental units \((\xi,\eta)\) may be of three different types, viz. \((\xi,\eta)= (\varepsilon, E)\) or \((\sqrt\varepsilon, E)\) or \((\varepsilon, \sqrt{\varepsilon})\), \(\varepsilon>1\) denoting the fundamental unit in the single quadratic subfield and \(E>1\), of degree four, the fundamental unit of the units in \(K(\alpha)\) with relative norm 1. It is proved that all three types occur in an infinite number of fields \(K(\alpha)\). The proofs are elementary.
scientific article; zbMATH DE number 3174539
scientific article