2021/05/27 by Ján Mináč, Minac, Jan, Andrew J. Schultz +3
Mathematics · #12F10 #16D70 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2105.13216
openalex publication_date 2021/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be prime, and n,m ∈ ℕ. When K/F is a cyclic extension of degree pn, we determine the ℤ/pmℤ[Gal(K/F)]-module structure of K^×/K× pm. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of Gal(K/F). For fixed values of m and n, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when m,n>1 the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.