Computing class fields via the Artin map (Q2719081)

To explicitly construct a class field \(F\) over \(E\), we append to \(E\) a sufficiently large root of unity to obtain a Kummer extension \(G\supseteq F\). For \(G/E\) a minimal number of radicals is found by using the Artin map to select radicands from \(S\)-units by the map into the identity, likewise for the selection of \(F\subseteq G\). An illustration even for the Hilbert class field \(E = \mathbb{Q}(\sqrt{10})\) is not trivial with modulus 1, class group \(C_2 =\langle 2_1\rangle\), \(2_1 = (2,\sqrt{10})\). By Hasse's Theorem [\textit{H. Hasse}, Vorlesungen über Klassenkörpertheorie, Physica-Verlag, Würzburg (1967; Zbl 0148.28005) pp. 232-233], \(G\) must allow for the new modulus \(2^5_1\infty_1\infty_2\) with \(\{3 + \sqrt{10}, -1,2\}\) as \(S\)-units in \(E\). The corresponding ray class is \(C^2_2\times C_4\) with three generating ideals \((1\pm 4\sqrt{10})\) (and prime factors \(\{3_1,3_2,53_1,53_2\})\) and \((5 + \sqrt{10})\) (ignored since it is in the unit ray class). Under the Artin mapping of the primes on all combinations of the \(S\)-units, only 2 is invariant and the class field is predictably \(E(\sqrt{2})\). The more difficult illustration with another modulus, e.g., \(9\infty_1\) with class group \(C_3\times C_4\) (and two Kummer extensions) is too long to summarize. Also a table of ray classfields for \(\mathbb{Q}(x)\), \(x^4 + 3x^3 + 2x + 1\) and various moduli up to 18 is shown.NEWLINENEWLINENEWLINEMuch remains to be done in simplifying or standardizing the results. [See \textit{H. Cohen}, Advanced topics in computational number theory, Springer Graduate Texts 193 (1999; Zbl 0977.11056), Chapter 5, and \textit{M. Daberkow} and \textit{M. E. Pohst}, J. Number Theory 69, 213-220 (1998; Zbl 0910.11058)].