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Galois module structure of the units modulo \(p^m\) of cyclic extensions of degree \(p^n\)

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A physical law must possess mathematical beauty.

Paul Dirac.

Abstract

Let p be prime, and \(n,m \in \mathbb {N}\). When K/F is a cyclic extension of degree \(p^n\), we determine the \(\mathbb {Z}/p^m\mathbb {Z}[\text {Gal}(K/F)]\)-module structure of \(K^\times /K^{\times p^m}\). With at most one exception, each indecomposable summand is cyclic and free over some quotient group of \(\text {Gal}(K/F)\). 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.

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Notes

  1. \({}^1\) no pun intended.

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Acknowledgements

We gratefully acknowledge discussions and collaborations with our friends and colleagues L. Bary-Soroker, S. Chebolu, F. Chemotti, I. Efrat, A. Eimer, J. Gärtner, S. Gille, P. Guillot, L. Heller, D. Hoffmann, J. Labute, C. Maire, D. Neftin, N.D. Tan, A. Topaz, R. Vakil and K. Wickelgren which have influenced our work in this and related papers. We would also like to thank the anonymous referee who provided valuable feedback that improved the quality and exposition of this manuscript.

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The first author is partially supported by the Natural Sciences and Engineering Research Council of Canada Grant R0370A01. He also gratefully acknowledges the Faculty of Science Distinguished Research Professorship, Western Science, in years 2004/2005 and 2020/2021. The second author was partially supported by 2017–2019 Wellesley College Faculty Awards. The third author was supported in part by National Security Agency grant MDA904-02-1-0061.

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  1. Department of Mathematics, Western University, London, ON, N6A 5B7, Canada

    Ján Mináč

  2. Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA, 02481, USA

    Andrew Schultz

  3. Office of the President, Carthage College, 2001 Alford Park Drive, Kenosha, WI, 53140, USA

    John Swallow

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The first author is partially supported by the Natural Sciences and Engineering Research Council of Canada grant R0370A01. He also gratefully acknowledges the Faculty of Science Distinguished Research Professorship, Western Science, in years 2004/2005 and 2020/2021. The second author was partially supported by 2017–2019 Wellesley College Faculty Awards. The third author was supported in part by National Security Agency grant MDA904-02-1-0061.

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Mináč, J., Schultz, A. & Swallow, J. Galois module structure of the units modulo \(p^m\) of cyclic extensions of degree \(p^n\). manuscripta math. 171, 295–345 (2023). https://doi.org/10.1007/s00229-022-01385-z

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