Let \(K\) be a field, \(\overline{K}/K\) the separable closure and \(G_K\) the absolute Galois group of \(K\), \(l,p\) are fixed primes.
Definition An \(l\)-adic representation of \(G_K\) is a finite dimensional \(\mathbb{Q}_l\)-vector space \(V\) with a continuous action of \(G_K\), \(\rho: G_K\to \mathrm{GL}(V)\cong \mathrm{GL}_d(\mathbb{Q}_l)\), \(d=\dim V\).
Since \(G_K\) is compact, we can always choose an \(\mathbb{Z}_l\)-lattice \(T\subset V\) that is \(G_K\)-stable, i.e. \(\rho(G_K)T\subset T\). Then we can define the \(l\)-adic Galois representation \(V_{\mathbb{Q}_l}:=T\otimes_{\mathbb{Z}_l}\mathbb{Q}_l\).