Galois representation

Remarks

Equivalently, it is a continuous action of $G_K$ on a free $A$-module of rank $n$, by $A$-linear maps. Two representations are isomorphic if they are conjugate by a matrix of $\mathrm{GL}_n(A)$.

Example: let $\ell$ be a prime different from the characteristic of $K$. The $\ell$-adic cyclotomic character $\chi_\ell \colon G_K \to \Z_\ell^\times = \mathrm{GL}_1(\Z_\ell)$ is given by $\sigma(\zeta) = \zeta^{\chi_\ell(\sigma) \bmod \ell^k}$. This holds for all $k$ and every root of unity $\zeta$ of order dividing $\ell^k$. It is a homomorphism, and it is continuous because its reduction modulo $\ell^k$ factors through the finite quotient $\Gal(K(\mu_{\ell^k})/K)$. Another example: the Tate module $T_\ell(E) = \varprojlim E[\ell^k]$ of an elliptic curve $E$ over $K$ gives a $2$-dimensional representation over $\Z_\ell$.

Non-example: the constant map $G_K \to \mathrm{GL}_n(A)$ with value $-I$, for $A$ of characteristic not $2$, is continuous but sends the identity to $-I \ne I$, so it is not a homomorphism.

With $A = \C$ and its usual topology, every continuous representation has finite image; these are the Artin representations. For $K$ a number field, a Galois representation is unramified at a finite place $v$ if it is trivial on the inertia group at $v$. Then the image of a Frobenius element at $v$ is defined up to conjugacy.