Nontrivial Galois module structure of cyclotomic fields

2002/01/17 by Marc Conrad, Conrad, Marc, Daniel R. Replogle +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.math/0201322

arxiv created 2002/01/17 · openalex publication_date 2002/01/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We say a tame Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that \CalO_L is a free \CalO_K[G]-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame Galois field extension of degree l so that L/K has nontrivial Galois module structure. However, the proof does not directly yield specific primes l for a given algebraic number field K. For K any cyclotomic field we find an explicit l so that there is a tame degree l extension L/K with nontrivial Galois module structure.

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