A *Galois module* is a $G$-module for a [[Galois group]] $G$; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure. The category of $G$-modules is equivalent to the category of [[module|modules]] over the [[group algebra|group ring]] $\mathbb{Z}]G]$. As always is the case, a group representation $G\times A\to A$ can equivalently be written as $G\to Aut(A)$. This is why Galois modules are frequently called *Galois representations*. ## Properties +-- {: .num_proposition} ###### Proposition Let $K\hookrightarrow L$ be a [[Galois extension]] of a number field $K$. Then the ring of integers $O_L$ of this extension is a [[Galois module]] of $Gal(K/hookrightarrow L)$. (see also [[Hilbert-Speiser theorem]]) =-- ## Examples +-- {: .num_example} ###### Example ($l$-adic representation) Let $l$ be a prime number. Let $Gal(k\hookrightarrow \overline k)$ be the [[absolute Galois group]] of a [[number field]] $k$. Then a morphism of groups $$Gal(k\hookrightarrow \overline k)\to Aut (M)$$ is called an *$l$-adic representation of $Gal(k\hookrightarrow \overline k)$. Here $M$ is either a unite dimensional [[vector space]] over the algebraic closure $\overline \mathbb{Q}_l$ or a finitely generated module over the [[integral closure]] $\overline \mathbb{Z}_l$. In particular the $l$-adic Tate-module is of this kind. =-- +-- {: .num_example} ###### Example ($l$-adic [[Tate module]]) Let $l$ be a prime number. Let $A$ be an abelian group. The *$l$-adic Tate module* is defined to be the limit $$T_l(A)=lim_n \;ker (l^n)$$ i.e. it is the [[directed limit|limit over the directed diagram]] $ker(p^{n+1})\to ker(p^n)$. Here the [[kernel]] $ker(p^n)$ of the multiplication-with-$p^n$ map $p^n:A\to A$ is called $p^n$-[[torsion]] of $A$. =-- +-- {: .num_example} ###### Example (*the* Tate-module) Let $k_S$ denote the separable closure of $k$. Let $A$ be the group of [[root of unity|roots of unity]] of $k_s$ in $k$. Then the $l$-adic Tate-module of the absolute Galois group $Gal(k\hookrightarrow k_s)$ is called *the $l$-adic Tate module of $k$* or the *$l$-adic cyclotomic character of $k$. It is equivalently the Tate-module of the [[multiplicative group scheme]] $\mu_k$. The Tate-module is endowed with the structure of a $\mathbb{Z}$-module by $z(a_n)_n=((z\; modulo\; p^n)a_n)_n$. =-- +-- {: .num_example} ###### Example ($l$-adic [[Tate module]] of an abelian variety) Let $l$ be a prime number. Let $G$ be an [[abelian variety]] over a field $k$. Let $k_s$ denote the separable closure of $k$. The $k_s$-valued points of $G$ assemble to an abelian group. Then there are classical results on the [[rank]] of the Tate-module $T_l(G)$: For example if the characteristic of $k$ is a prime number $p\neq l$ we have that $T_l(G)$ is a free $\mathbb{Z}_l$ module of rank $2dim(G)$. A special case of the [[Tate conjecture]] can be formulated via Tate-modules: Let $k$ be finitely generated over its prime field of characteristic $p\neq l$. Let $A,B$ be two abelian varieties over $k$. Then the conjecture states that $$hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))$$ If $k$ is a finite field or a number field the conjecture is true. =-- ## Galois modules and quasi finite fields