English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

is the Hilbert 2-class field tower of k. We say that the tower is finite if kn = kn+1 for some n, and infinite otherwise. We define the 2-rank of Ck as the dimension of the elementary abelian 2-group Ck/C k viewed as a vector space over F2: rank2(Ck) = dimF2(Ck/C 2 k), where F2 is the finite field with two elements. We define the 4-rank of Ck by rank4(Ck) = rank2(C k) = dimF2(C 2 k/C 4 k). Assume k is an imaginary quadratic number field. It is well known that if rank2(Ck) ≥ 5, then the Hilbert 2-class field tower of k is infinite [5]. In the case where rank2(Ck) = 2 or 3, the Hilbert 2-class field tower of k may be finite ([9], [10]), and if rank2(Ck) = 1 then the Hilbert 2-class field tower of k is finite of length 1. It has been conjectured that if rank2(Ck) = 4, then k has infinite Hilbert 2-class field tower [10]. We mention that Hajir proved that if Ck contains a subgroup isomorphic to Z/4Z× Z/4Z× Z/4Z, then k has infinite Hilbert 2-class field tower ([6], [7]). Now suppose that rank2(Ck) = 4 and the discriminant of k is divisible by exactly one negative prime discriminant. In [2], under some conditions on the 4-rank of Ck and the Kronecker symbols of the primes dividing the discriminant of k, the author proves that k has infinite Hilbert 2-class field tower. Y. Sueyoshi proves the same result under some conditions on the Rédei matrix [14].

Infinite Hilbert 2-class field tower of quadratic number fields