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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]] with separable closure $K^{\\mathrm{sep}}$, and let $G_K = \\Gal(K^{\\mathrm{sep}}/K)$ be its absolute [[Galois group|0XTY86BZKY]] with the Krull topology, a profinite group.\nLet $A$ be a topological commutative ring and let $n \\ge 1$.\nTypical choices of $A$ are $\\C$, a finite field, $\\Z_\\ell$, or a finite extension of $\\Q_\\ell$.\nGive the [[general linear group|0FCHE902D7]] $\\mathrm{GL}_n(A)$ the topology induced from $A^{n^2}$.\nA *Galois representation* of $K$ of dimension $n$ over $A$ is a [[continuous|0X3WGX95FE]] [[group homomorphism|0JYTG5K9Y4]] $\\rho \\colon G_K \\to \\mathrm{GL}_n(A)$.\n\n## Remarks\n\nEquivalently, it is a continuous action of $G_K$ on a free $A$-module of rank $n$, by $A$-linear maps.\nTwo representations are isomorphic if they are conjugate by a matrix of $\\mathrm{GL}_n(A)$.\n\nExample: let $\\ell$ be a prime different from the characteristic of $K$.\nThe $\\ell$-adic cyclotomic character $\\chi_\\ell \\colon G_K \\to \\Z_\\ell^\\times = \\mathrm{GL}_1(\\Z_\\ell)$ is given by $\\sigma(\\zeta) = \\zeta^{\\chi_\\ell(\\sigma) \\bmod \\ell^k}$.\nThis holds for all $k$ and every root of unity $\\zeta$ of order dividing $\\ell^k$.\nIt is a homomorphism, and it is continuous because its reduction modulo $\\ell^k$ factors through the finite quotient $\\Gal(K(\\mu_{\\ell^k})/K)$.\nAnother example: the Tate module $T_\\ell(E) = \\varprojlim E[\\ell^k]$ of an elliptic curve $E$ over $K$ gives a $2$-dimensional representation over $\\Z_\\ell$.\n\nNon-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.\n\nWith $A = \\C$ and its usual topology, every continuous representation has finite image; these are the Artin representations.\nFor $K$ a number field, a Galois representation is unramified at a finite place $v$ if it is trivial on the inertia group at $v$.\nThen the image of a Frobenius element at $v$ is defined up to conjugacy.\n",
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