2002/11/13 by Timothy Kohl, Kohl, Timothy, Daniel R. Replogle +2
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.math/0211468
arxiv created 2002/11/13 · openalex publication_date 2002/11/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Km=ℚ(ζm) where ζm is a primitive mth root of unity. Let p>2 be prime and let Cp denote the group of order p. The ring of algebraic integers of Km is \CalOm=ℤ[ζm]. Let Λm,p denote the order \CalOm[Cp] in the algebra Km[Cp]. Consider the kernel group D(Λm,p) and the Swan subgroup T(Λm,p). If (p,m)=1 these two subgroups of the class group coincide. Restricting to when there is a rational prime p that is prime in \CalOm requires m=4 or qn where q>2 is prime. For each such m, 3 ≤ m ≤ 100, we give such a prime, and show that one may compute T(Λm,p) as a quotient of the group of units of a finite field. When hmp+=1 we give exact values for |T(Λm,p)|, and for other cases we provide an upper bound. We explore the Galois module theoretic implications of these results.