# Galois module

A **Galois module** is an abelian group, most often the ring of integers of a field extension, on which a [Galois group](https://www.edgepedia.org/galois-group) acts compatibly with the group structure; equivalently, an action of a Galois group \( G \) on \( M \) is a homomorphism \( G \to \mathrm{Aut}(M) \)<sup>[1](https://ncatlab.org/nlab/show/Galois%20module)</sup>. The term also designates the branch of algebraic number theory, Galois module structure theory, that studies how the ring of integers \( O_L \) of a finite [Galois extension](https://www.edgepedia.org/galois-extension) \( L/K \) sits as a module over the group ring \( O_K[G] \), where \( G = \mathrm{Gal}(L/K) \)<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. When \( G \) acts on a module \( M \) that also carries an action of a commutative ring \( K \) commuting with \( G \), \( M \) becomes a \( KG \)-module by extending the group action linearly<sup>[3](https://encyclopediaofmath.org/wiki/Module)</sup>.

| Key fact | Detail |
|---|---|
| Definition | An abelian group with an action of a Galois group \( G \), equivalently \( G \to \mathrm{Aut}(M) \)<sup>[1](https://ncatlab.org/nlab/show/Galois%20module)</sup> |
| Central object | \( O_L \) as an \( O_K[G] \)-module for a finite Galois extension \( L/K \)<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup> |
| Noether's criterion | \( O_L \) is locally free over \( O_K[G] \) if and only if \( L/K \) is tamely ramified<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup> |
| Normal integral basis | \( O_L \) is free of rank 1 over \( O_K[G] \), generated by the Galois conjugates of one element<sup>[4](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/lucchini.pdf)</sup> |
| Hilbert–Speiser theorem | Every tame abelian extension of \( \mathbb{Q} \) has a normal integral basis; \( \mathbb{Q} \) is the only Hilbert–Speiser number field<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup><sup> • </sup><sup>[5](https://doi.org/10.1007/s10231-019-00852-x)</sup> |
| Fröhlich conjecture | Taylor proved in 1981 that for tame \( L/\mathbb{Q} \), the class of \( O_L \) in \( \mathrm{Cl}(\mathbb{Z}[G]) \) is determined by Artin root numbers<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup> |
| Wild case | For wildly ramified extensions, \( O_L \) is studied over associated orders and Hopf orders, using tools such as Galois scaffolds<sup>[6](https://researchwith.stevens.edu/en/publications/galois-scaffolds-for-extraspecial-p-extensions-in-characteristic-/)</sup> |

## The module structure of rings of integers

For a finite Galois extension \( L/K \) of number fields or local fields with rings of integers \( O_K \subset O_L \) and Galois group \( G \), the action of \( G \) on \( L \) restricts to \( O_L \), giving \( O_L \) a natural \( O_K[G] \)-module structure<sup>[4](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/lucchini.pdf)</sup>. The basic question of the theory is to describe this module: whether it is free, projective, or locally free, and when it is not free, to compute the class measuring the failure.

The strongest possible statement is the existence of a **normal integral basis**: an element \( \alpha \in L \) whose Galois conjugates form an \( O_K \)-basis of \( O_L \), which is equivalent to \( O_L \) being a free \( O_K[G] \)-module of rank 1<sup>[4](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/lucchini.pdf)</sup>. This is the integral analog of the normal basis theorem, which guarantees a \( K \)-basis of conjugates for any finite Galois extension<sup>[7](https://www.math.ucla.edu/~rse/algebra_book.pdf)</sup>.

## Noether's criterion

Ramification controls the answer. A theorem presented as the starting point of the subject states, for a [Dedekind domain](https://www.edgepedia.org/dedekind-domain) \( A \) with field of fractions \( K \) and integral closure \( B \) in a finite Galois extension \( L/K \), the equivalence of three conditions: \( B \) is a projective \( A[G] \)-module; the trace map satisfies \( \mathrm{Tr}_{L/K}(B) = A \); and \( L/K \) is at most tamely ramified<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. The equivalence of the trace condition with tameness follows from the characterization of the different \( \mathcal{D}_{L/K} \), since \( \mathrm{Tr}_{L/K}(B) \neq A \) exactly when \( \mathcal{D}_{L/K} \) is contained in \( \mathfrak{p}B \) for some prime \( \mathfrak{p} \) of \( A \)<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>.

Combining this with a theorem of Swan, that every finitely generated projective \( O_K[G] \)-module is locally free, yields **Noether's criterion**: \( O_L \) is locally free over \( O_K[G] \) if and only if \( L/K \) is tamely ramified<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. Part of the criterion goes back to [Andreas Speiser](https://www.edgepedia.org/andreas-speiser), who proved the necessary condition, and the criterion is presented as the basic starting point of Galois module structure theory<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. [Noether's theorem](https://www.edgepedia.org/noethers-theorem) in its local form gives the same equivalence for freeness of rank one over \( O_K[G] \)<sup>[8](https://people.cs.dm.unipi.it/stefanello/bristol2021.pdf)</sup>.

## The class group and the Fröhlich conjecture

When \( L/K \) is tame, \( O_L \) determines an element in the locally free class group \( \mathrm{Cl}(O_K[G]) \), defined as the kernel of the rank map from the Grothendieck group \( K_0(O_K[G]) \) to \( \mathbb{Z} \)<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. This group is a finite abelian group, and locally free modules of rank at least 2 satisfy cancellation, so the class of \( O_L \) records precisely whether \( O_L \) is free<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>.

**Over \( \mathbb{Q} \).** For tame abelian extensions \( L/\mathbb{Q} \), the Hilbert–Speiser theorem gives a normal [integral basis](https://www.edgepedia.org/integral-basis); originally proved for abelian extensions of degree prime to the discriminant, it was extended to all tame abelian extensions by Leopoldt<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. The proof uses the [Kronecker–Weber theorem](https://www.edgepedia.org/kronecker-weber-theorem): \( L \) lies in a cyclotomic field \( \mathbb{Q}(e^{2i\pi/n}) \) with \( n \) squarefree, and the trace of \( e^{2i\pi/n} \) generates a normal integral basis<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>.

The non-abelian case was investigated through the 1970s by Armitage, Cassou-Noguès, Cougnard, Fröhlich, Martinet, Queyrut, and Taylor. Martinet first proved freeness for dihedral groups of order \( 2p \), \( p \) an odd prime, but in 1971 constructed tame extensions of \( \mathbb{Q} \) with quaternion Galois group of order 8 whose rings of integers are not free over \( \mathbb{Z}[G] \), the first known counterexample to the existence of normal integral bases<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. This motivated the conjecture of [Albrecht Fröhlich](https://www.edgepedia.org/albrecht-frohlich), which determines the class of \( O_L \) in \( \mathrm{Cl}(\mathbb{Z}[G]) \) in terms of an analytic invariant, the root number class \( t(W_{L/\mathbb{Q}}) \) built from Artin root numbers of symplectic characters<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>.

Taylor proved the conjecture in 1981, using a generalization of Lagrange resolvents and Galois Gauss sums to non-abelian characters, a logarithm for local group rings, and Fröhlich's Hom-description of the class group<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. The theorem states that for a tame extension \( L/\mathbb{Q} \), \( O_L \oplus O_L \) is stably free, hence free of rank 2, and the only obstruction to \( [O_L] = 1 \) is the signs of the Artin root numbers of symplectic characters; in particular \( O_L \) is free whenever \( G \) has no irreducible symplectic character<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>. The conjecture also generated new tame extensions of \( \mathbb{Q} \) without normal integral bases, including quaternion examples constructed by Cougnard<sup>[2](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)</sup>.

## Hilbert–Speiser and Leopoldt fields

A number field \( K \) is called **Hilbert–Speiser** if all of its tame abelian extensions admit normal integral bases; \( \mathbb{Q} \) is the only such field<sup>[5](https://doi.org/10.1007/s10231-019-00852-x)</sup>. Weakening freeness to the existence of a weak normal integral basis defines \( G \)-Leopoldt fields, for which the classification for a fixed Galois group \( G \) remains incomplete, with many results limiting the class of such fields, sometimes to a finite list<sup>[5](https://doi.org/10.1007/s10231-019-00852-x)</sup>.

## Wildly ramified extensions and associated orders

Noether's criterion shows that another approach is needed when the extension is wildly ramified, since \( O_L \) is then not projective over \( O_K[G] \)<sup>[4](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/lucchini.pdf)</sup>. The natural substitute is the **associated order** \( A_{L/K} \), the suborder of \( K[G] \) consisting of elements preserving \( O_L \), and the question becomes whether \( O_L \) is free of rank 1 over \( A_{L/K} \)<sup>[6](https://researchwith.stevens.edu/en/publications/galois-scaffolds-for-extraspecial-p-extensions-in-characteristic-/)</sup>. Recent work gives sufficient conditions for totally ramified extensions with extraspecial \( p \)-group Galois group over a \( p \)-adic field of residue characteristic \( p > 2 \) to admit a Galois scaffold, which yields freeness over the associated order and, under stricter conditions, makes \( A_{L/K} \) a Hopf order in \( K[G] \)<sup>[6](https://researchwith.stevens.edu/en/publications/galois-scaffolds-for-extraspecial-p-extensions-in-characteristic-/)</sup>.

A **Galois scaffold** adapts the normal basis to the valuation of the extension field<sup>[9](https://arxiv.org/abs/1106.3577)</sup>. In the general framework of \( A \)-scaffolds for totally ramified \( p \)-extensions of complete local fields, a necessary and sufficient condition involving only numerical parameters, the ramification breaks in the Galois case, determines whether a fractional ideal is free over its associated order, and determines the number of generators needed when it is not<sup>[10](https://www.numdam.org/articles/10.5802/aif.3182/)</sup>.

## Extensions beyond rings of integers

The Galois module viewpoint applies to other arithmetic objects. The Encyclopedia of Mathematics lists, for a number field \( K \), the additive group of its ring of integers, the group of units, the group of divisors, and the divisor class group among the \( G \)-modules arising from a Galois group of \( K \)<sup>[3](https://encyclopediaofmath.org/wiki/Module)</sup>. In geometric settings, Nakajima studied the \( k[G] \)-module structure of the cohomology groups \( H^i(X, \mathcal{F}) \) for tamely ramified Galois coverings \( f : X \to Y \) of projective varieties, showing for instance that \( H^n(X, \mathcal{F}) \) is \( k[G] \)-projective when the other cohomology vanishes<sup>[11](https://www.sciencedirect.com/science/article/pii/0022314X86900326)</sup>. For Galois covers of curves over \( \mathbb{F}_p \), the \( p \)-adic valuation of the epsilon constants in functional equations of Artin L-functions is related to an equivariant [Euler characteristic](https://www.edgepedia.org/euler-characteristic), generalizing a result of Chinburg from the tamely to the weakly ramified case<sup>[12](https://ems.press/journals/dm/articles/8965609)</sup>. The action of a Galois group on the Mordell–[Weil group](https://www.edgepedia.org/weil-group) of an elliptic curve is studied similarly: for an odd prime \( p \), the \( p \)-adic completion of \( E(K) \) is analyzed as a \( \mathbb{Z}_p[G] \)-module using data over the base field and local completions<sup>[13](https://arxiv.org/abs/2306.13365)</sup>.

## Hopf–Galois structures

Hopf–[Galois theory](https://www.edgepedia.org/galois-theory) widens the framework by replacing the group algebra \( K[G] \) with other Hopf algebras acting on \( L \). This is particularly useful for extensions that are not Galois, where classical techniques do not apply: for a tamely ramified radical extension of prime degree \( p \) of number fields, the ring of integers is locally free over its associated order in the unique Hopf–Galois structure, with an explicit criterion for freeness<sup>[14](https://www.sciencedirect.com/science/article/pii/S0022404919302385)</sup>. Nonclassical Hopf–Galois module structures have also been studied for rings of integers in \( p \)-adic and number field extensions that are at most tamely ramified<sup>[15](https://arxiv.org/abs/1001.1639)</sup>.

## References

1. [Galois module, nLab](https://ncatlab.org/nlab/show/Galois%20module)
2. [Lara Thomas, On the Galois module structure of extensions of local fields, Publications mathématiques de Besançon (2010)](https://www.numdam.org/item/10.5802/pmb.a-131.pdf)
3. [Module, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Module)
4. [Andrea Lucchini, PhD thesis on Galois module structure of local fields, ALGANT](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/lucchini.pdf)
5. [Tame Galois module structure revisited, Annali di Matematica Pura ed Applicata (Springer)](https://doi.org/10.1007/s10231-019-00852-x)
6. [Kevin Keating and Paul Schwartz, Galois scaffolds for extraspecial p-extensions in characteristic 0, Journal of Number Theory 278 (2026)](https://researchwith.stevens.edu/en/publications/galois-scaffolds-for-extraspecial-p-extensions-in-characteristic-/)
7. [Ravi Shankar, Algebra, UCLA lecture notes (Normal Basis Theorem)](https://www.math.ucla.edu/~rse/algebra_book.pdf)
8. [Andrea Sivacara Stefanello, Local Galois module theory: an overview, Università di Pisa](https://people.cs.dm.unipi.it/stefanello/bristol2021.pdf)
9. [Byott, Elder, Galois scaffolds and Galois module structure in extensions of characteristic p local fields of degree p², arXiv](https://arxiv.org/abs/1106.3577)
10. [Byott, Childs, Elder, Scaffolds and generalized integral Galois module structure, Annales de l'Institut Fourier 68 (2018)](https://www.numdam.org/articles/10.5802/aif.3182/)
11. [Shōichi Nakajima, Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties, Journal of Number Theory 22 (1986)](https://www.sciencedirect.com/science/article/pii/0022314X86900326)
12. [Fischbacher-Weitz, Köck, Galois-Module Theory for Wildly Ramified Covers of Curves over Finite Fields, Documenta Mathematica 24 (2019)](https://ems.press/journals/dm/articles/8965609)
13. [Vavasour, Wuthrich, Mordell-Weil groups as Galois modules, arXiv (2023)](https://arxiv.org/abs/2306.13365)
14. [Hopf-Galois module structure of tamely ramified radical extensions of prime degree, Journal of Pure and Applied Algebra 224 (2020)](https://www.sciencedirect.com/science/article/pii/S0022404919302385)
15. [Towards a Generalisation of Noether's Theorem to Nonclassical Hopf-Galois Structures, arXiv](https://arxiv.org/abs/1001.1639)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Field and Galois theory › Galois cohomology*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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