Maximal unramified extensions of imaginary quadratic number fields of small conductors (Q5906914)

This paper deals with the structure of the Galois groups \(\text{Gal} (K_{ur}/K)\) of the maximal unramified extensions \(K_{ur}\) of imaginary quadratic number fields \(K\) of conductors \(\leq 420\) \((\leq 719\) under the Generalized Riemann Hypothesis). The author shows that for all such \(K\), \(K_{ur}\) is precisely the top of the class field tower of \(K\) and the length of the tower is at most three. The techniques to establish this result rest upon the inequality \([K_{ur}: K_1] <60\), where \(K_1\) is the top of the class field tower of \(K\), and the number 60 is significant as the smallest order of a nonsolvable group, namely \(A_5\) (the alternating group of degree 5). The above inequality \([K_{ur}: K_1] <60\) is obtained from lower bounds on the root discriminants of totally imaginary number fields, based upon the tables of Diaz and Diaz (unconditionally) and Odlyzko (assuming the Generalized Riemann Hypothesis). In order to compute \(\text{Gal} (K_{ur}/K)\), extensive use is made of \(p\)-class field towers, class number formulas, genus field class structure, the action of Galois groups on class groups, group tables of order \(\leq 64\), and computer calculations. Finally, the author includes a number of examples, demonstrating the use of his techniques for imaginary quadratic number fields \(K=\mathbb{Q} (\sqrt {-d})\) with \(723 \leq| d| <1000\), and makes the hypothesis that \(K_{ur} =K_1\) holds for all \(K\) with \(| d| <1507\), based upon his expectation that \(\mathbb{Q} (\sqrt {-1507})\) is the first \(K\) having an unramified nonsolvable Galois extension. In a supplement available from the author that includes some minor corrections to the paper, the table is improved upon and extended to real quadratic number fields as well.