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Computation of Galois groups associated to the 2-class towers of some quadratic fields. (English) Zbl 1039.11091

Let \(k\) be a complex quadratic number field; the \(2\)-class field \(k^1\) of \(k\) is the maximal abelian unramified \(2\)-extension of \(k\), and its Galois group is isomorphic to the \(2\)-class group of \(k\). Iterating this construction gives a tower of field extensions \(k \subseteq k^1 \subseteq k^2 \subseteq \cdots\) called the \(2\)-class field tower. It has been known for a long time that the \(2\)-class field tower of \(k\) is finite if its \(2\)-class group is cyclic, and by a result of Golod and Shafarevich [see Roquette’s article On class field towers in J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press (1967; Zbl 0153.07403)], the \(2\)-class field tower is infinite if the \(2\)-class group has rank at least \(5\). In recent years, the open cases where the \(2\)-rank equals \(2\), \(3\) or \(4\) have been studied in particular by E. Benjamin, F. Lemmermeyer and C. Snyder, see e.g. [J. Number Theory 103, No. 1, 38–70 (2003; Zbl 1045.11077)].
In this article, the author investigates the \(2\)-class field tower of the smallest open examples, namely the fields with discriminant \(d = -445, -1015, -1595\), and \(-2379\). Calculations combining group theory with the computation of certain class groups then show that the \(2\)-class field tower is finite in each of these cases.
Consider e.g. the field \(k\) with discriminant \(d = -2379\). Its \(2\)-class group has type \([4,4]\), hence the Galois group \(G\) of its \(2\)-class field tower satisfies \(G/G' \simeq [4,4]\). The subgroups of index \(2\) in \(G\) must have abelianization isomorphic to the \(2\)-class groups of the unramified quadratic extensions of \(k\), which can be computed easily. Using similar results for the subfields of some octic unramified extension of \(k\) it is then possible to give a finite list of possible groups \(G\); all of these groups happen to have the same cardinality \(\# G = 2^{10}\) and satisfy \(G/G' \simeq [4,4]\) and \(G'/G'' \simeq [2,4,16]\). In particular, the \(2\)-class field tower of \(k\) terminates after the second step, and the \(2\)-class group of the \(2\)-class field of \(k\) has type \([2,4,16]\).

MSC:

11Y40 Algebraic number theory computations
11R37 Class field theory
11R11 Quadratic extensions

Software:

KANT/KASH; Magma
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Full Text: DOI arXiv

References:

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