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In mathematics, a **Galois module** is a *G*-module, with *G* being the Galois group of some extension of fields. The term **Galois representation** is frequently used when the *G*-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for *G*-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

- Given a field
*K*, the multiplicative group (*K*)^{s}^{×}of a separable closure of*K*is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of*K*(by Hilbert's theorem 90, its first cohomology group is zero). - If
*X*is a smooth proper scheme over a field*K*then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of*K*.

Let *K* be a valued field (with valuation denoted *v*) and let *L*/*K* be a finite Galois extension with Galois group *G*. For an extension *w* of *v* to *L*, let *I _{w}* denote its inertia group. A Galois module ρ :

In classical algebraic number theory, let *L* be a Galois extension of a field *K*, and let *G* be the corresponding Galois group. Then the ring *O*_{L} of algebraic integers of *L* can be considered as an *O*_{K}[*G*]-module, and one can ask what its structure is. This is an arithmetic question, in that by the normal basis theorem one knows that *L* is a free *K*[*G*]-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a **normal integral basis**, i.e. of α in *O*_{L} such that its conjugate elements under *G* give a free basis for *O*_{L} over *O*_{K}. This is an interesting question even (perhaps especially) when *K* is the rational number field **Q**.

For example, if *L* = **Q**(√−3), is there a normal integral basis? The answer is yes, as one sees by identifying it with **Q**(*ζ*) where

*ζ*= exp(2π*i*/3).

In fact all the subfields of the cyclotomic fields for *p*-th roots of unity for *p* a *prime number* have normal integral bases (over **Z**), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem). On the other hand, the Gaussian field does not. This is an example of a *necessary* condition found by Emmy Noether (*perhaps known earlier?*). What matters here is *tame* ramification. In terms of the discriminant *D* of *L*, and taking still *K* = **Q**, no prime *p* must divide *D* to the power *p*. Then Noether's theorem states that tame ramification is necessary and sufficient for *O _{L}* to be a projective module over

A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into a cyclotomic field.^{[1]}

Many objects that arise in number theory are naturally Galois representations. For example, if *L* is a Galois extension of a number field *K*, the ring of integers *O _{L}* of

There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the ℓ-adic Tate modules of abelian varieties.

Let *K* be a number field. Emil Artin introduced a class of Galois representations of the absolute Galois group *G _{K}* of

Because of the incompatibility of the profinite topology on *G _{K}* and the usual (Euclidean) topology on complex vector spaces, the image of an Artin representation is always finite.

Let ℓ be a prime number. An **ℓ-adic representation** of *G _{K}* is a continuous group homomorphism ρ :

Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of *G*_{Q} under the ℓ-adic cyclotomic character is . ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of **Q**_{ℓ} with **C** they can be identified with *bona fide* Artin representations.

These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.

There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name with different meanings. Some of these conditions include:

- Abelian representations. This means that the image of the Galois group in the representations is abelian.
- Absolutely irreducible representations. These remain irreducible over an algebraic closure of the field.
- Barsotti–Tate representations. These are similar to finite flat representations.
- Crystabelline representations
- Crystalline representations.
- de Rham representations.
- Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat group scheme.
- Good representations. These are related to the representations of elliptic curves with good reduction.
- Hodge–Tate representations.
- Irreducible representations. These are irreducible in the sense that the only subrepresentation is the whole space or zero.
- Minimally ramified representations.
- Modular representations. These are representations coming from a modular form, but can also refer to representations over fields of positive characteristic.
- Ordinary representations. These are related to the representations of elliptic curves with ordinary (non-supersingular) reduction. More precisely, they are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character ε on the submodule.
- Potentially
*something*representations. This means that the representations restricted to an open subgroup of finite index has some specified property. - Reducible representations. These have a proper non-zero sub-representation.
- Semistable representations. These are two dimensional representations related to the representations coming from semistable elliptic curves.
- Tamely ramified representations. These are trivial on the (first) ramification group.
- Trianguline representations.
- Unramified representations. These are trivial on the inertia group.
- Wildly ramified representations. These are non-trivial on the (first) ramification group.

If *K* is a local or global field, the theory of class formations attaches to *K* its Weil group *W _{K}*, a continuous group homomorphism φ :

where *C _{K}* is

An ℓ-adic representation of *W _{K}* is defined in the same way as for

Let *K* be a local field. Let *E* be a field of characteristic zero. A **Weil–Deligne representation** over *E* of *W _{K}* (or simply of

- a continuous group homomorphism
*r*:*W*→ Aut_{K}_{E}(*V*), where*V*is a finite-dimensional vector space over*E*equipped with the discrete topology, - a nilpotent endomorphism
*N*:*V*→*V*such that*r*(*w*)N*r*(*w*)^{−1}= ||*w*||*N*for all*w*∈*W*._{K}^{[2]}

These representations are the same as the representations over *E* of the Weil–Deligne group of *K*.

If the residue characteristic of *K* is different from ℓ, Grothendieck's ℓ-adic monodromy theorem sets up a bijection between ℓ-adic representations of *W _{K}* (over

**^**Fröhlich 1983, p. 8**^**Here ||*w*|| is given by*q**v*(*w*)*K*where*q*is the size of the residue field of_{K}*K*and*v*(*w*) is such that*w*is equivalent to the −*v*(*w*)th power of the (arithmetic) Frobenius of*W*._{K}

- Kudla, Stephen S. (1994), "The local Langlands correspondence: the non-archimedean case",
*Motives, Part 2*, Proc. Sympos. Pure Math., vol. 55, Providence, R.I.: Amer. Math. Soc., pp. 365–392, ISBN 978-0-8218-1635-6 - Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000),
*Cohomology of Number Fields*,*Grundlehren der Mathematischen Wissenschaften*, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001 - Tate, John (1979), "Number theoretic background",
*Automorphic forms, representations, and L-functions, Part 2*, Proc. Sympos. Pure Math., vol. 33, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 978-0-8218-1437-6

- Snaith, Victor P. (1994),
*Galois module structure*, Fields Institute monographs, Providence, RI: American Mathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042 - Fröhlich, Albrecht (1983),
*Galois module structure of algebraic integers*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 1, Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, ISBN 3-540-11920-5, Zbl 0501.12012