# Applications of representation theory and of explicit units to Leopoldt’s conjecture

## TL;DR

This paper explores how properties of number fields and their symmetries can help verify a long-standing mathematical conjecture called Leopoldt’s conjecture.

## Contribution

The paper introduces new implications between Leopoldt’s conjecture for intermediate fields and the base field, using representation theory and explicit unit constructions.

## Key findings

- Leopoldt’s conjecture at p for intermediate fields implies it for the full extension under certain group conditions.
- Relations between Leopoldt defects of intermediate fields are established.
- An infinite family of number fields is shown to satisfy Leopoldt’s conjecture for any finite set of primes.

## Abstract

Let L/K be a Galois extension of number fields and let \documentclass[12pt]{minimal}
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\begin{document}$$G=\textrm{Gal}(L/K)$$\end{document}G=Gal(L/K). We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt’s conjecture at p for certain proper intermediate fields of L/K implies Leopoldt’s conjecture at p for L. We also obtain relations between the Leopoldt defects of intermediate fields of L/K. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document}P, there exists an infinite family \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}F of totally real \documentclass[12pt]{minimal}
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\begin{document}$$S_{3}$$\end{document}S3-extensions of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Q}$$\end{document}Q such that Leopoldt’s conjecture for F at p holds for every \documentclass[12pt]{minimal}
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\begin{document}$$F \in \mathcal {F}$$\end{document}F∈F and \documentclass[12pt]{minimal}
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\begin{document}$$p \in \mathcal {P}$$\end{document}p∈P.