Infinite Hilbert 2-class field tower of quadratic number fields. (English) Zbl 1211.11127
Acta Arith. 145, No. 3, 267-272 (2010); acknowledgement of priority ibid. 167, No. 3, 299 (2015).
The author improves upon previous results to show that it \(k\) is an imaginary quadratic number field whose discriminant is divisible by at most one negative prime discriminant and the 2-rank of the class group is 4, then \(k\) has infinite Hilbert 2-class field tower. The author establishes this result by making use of the Golod and Shafarevich inequality, results from genus theory, and the ramification of primes. This result is significant as it is a further step in answering a question of Martinet, which is to determine if it is true that an imaginary quadratic number field whose class group has 2-rank 4, has infinite Hilbert 2-class field tower (it has been conjectured that this statement is true). Using similar techniques as above, the author also shows that a positive proportion of imaginary quadratic number fields whose class group has 2-rank 2 and 4-rank 1 have infinite Hilbert 2-class field tower.
Appendix, added January 07, 2015: I have recently learned that the second result mentioned in my review, which is that a positive proportion of imaginary quadratic number fields whose class group has 2-rank 2 and 4-rank 1 have infinite Hilbert 2-class field tower, was essentially proved in 1980 by B. Schmithals [Arch. Math. 34, 307–312 (1980; Zbl 0448.12008)]. The author refers to Schmithals’ 1980 paper for a related example, but does not acknowledge Schmithals’ proof of Satz 1 in this 1980 paper, which establishes that there are infinitely many imaginary quadratic number fields \(k\) whose discriminant is divisible by exactly three prime discriminants and have infinite 2-class field tower. It is well known (see for example L. Redei and H. Reichardt’s article [J. Reine Angew. Math. 170, 69–74 (1933; Zbl 0007.39602; JFM 59.0192.01)] the author refers to) that the assumptions of Satz 1 and the techniques used in the proof imply that the class group of k has 2-rank 2 and 4-rank 1, and consequently we see that this result by the author whose paper I reviewed is not a new result. For historical accuracy and respect for B. Schmithals, I therefore am including this appendix to my initial review.
Editorial remark: See the acknowledgement of priority in [ibid. 167, No. 3, 299–299 (2015; Zbl 1316.11103)].
Appendix, added January 07, 2015: I have recently learned that the second result mentioned in my review, which is that a positive proportion of imaginary quadratic number fields whose class group has 2-rank 2 and 4-rank 1 have infinite Hilbert 2-class field tower, was essentially proved in 1980 by B. Schmithals [Arch. Math. 34, 307–312 (1980; Zbl 0448.12008)]. The author refers to Schmithals’ 1980 paper for a related example, but does not acknowledge Schmithals’ proof of Satz 1 in this 1980 paper, which establishes that there are infinitely many imaginary quadratic number fields \(k\) whose discriminant is divisible by exactly three prime discriminants and have infinite 2-class field tower. It is well known (see for example L. Redei and H. Reichardt’s article [J. Reine Angew. Math. 170, 69–74 (1933; Zbl 0007.39602; JFM 59.0192.01)] the author refers to) that the assumptions of Satz 1 and the techniques used in the proof imply that the class group of k has 2-rank 2 and 4-rank 1, and consequently we see that this result by the author whose paper I reviewed is not a new result. For historical accuracy and respect for B. Schmithals, I therefore am including this appendix to my initial review.
Editorial remark: See the acknowledgement of priority in [ibid. 167, No. 3, 299–299 (2015; Zbl 1316.11103)].
Reviewer: Elliot Benjamin (Swanville)
