Infinite families of cyclotomic function fields with any prescribed class group rank
- 주제(키워드) Class group rank , Cyclotomic function field , Ideal class group , Kummer extension , Maximal real subfield
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 Elsevier B.V.
- 발행년도 2021
- 총서유형 Journal
- URI http://www.dcollection.net/handler/ewha/000000175522
- 본문언어 영어
- Published As http://dx.doi.org/10.1016/j.jpaa.2020.106658
초록/요약
We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k=Fq(T) whose class groups can have arbitrarily large ℓn-rank, where Fq is the finite field of prime power order q. We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k(Λ)+ of cyclotomic function fields k(Λ) whose ideal class groups have arbitrary ℓn-rank for n = 1, 2, and 3, where ℓ is a prime divisor of q−1. We also obtain a tower of cyclotomic function fields Ki whose maximal real subfields have ideal class groups of ℓn-ranks getting increased as the number of the finite places of k which are ramified in Ki get increased for i≥1. Our main idea is to use the Kummer extensions over k which are subfields of k(Λ)+, where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields k(Λ)+ of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k. We demonstrate our results by presenting some examples calculated by MAGMA at the end. © 2020 Elsevier B.V.
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