Let $G_{\Rats}=\Gal(\overline{\Rats}/\Rats)$ be the absolute Galois group of $\Rats$ . The purpose of this entry is to define, for every prime $p$ , a Galois representation:

$$\chi_p : G_{\Rats} \longrightarrow \Ints_p^\times$$

where $\Ints_p^\times$ is the group of units of $\Ints_p$ , the $p$ -adic integers. $\chi_p$ is a $\Ints_p^\times$ valued character, usually called the cyclotomic character of $G_{\Rats}$ , or the $p$ -adic cyclotomic Galois representation of $G_{\Rats}$ . Here is the construction:

For each $n\geq 1$ , let $\zeta_{p^n}$ be a primitive $p^n$ -th root of unity and let $K_n=\Rats(\zeta_{p^n})$ be the corresponding cyclotomic extension of $\Rats$ . By the basic theory of cyclotomic extensions, we know that

$$\Gal(K_n/\Rats)\cong (\Ints/p^n\Ints)^\times.$$

Moreover, the restriction map $\Gal(K_{n+1}/\Rats)\to \Gal(K_n/\Rats)$ is given by reduction modulo $p^n$ from $(\Ints/p^{n+1}\Ints)^\times$ to $(\Ints/p^n\Ints)^\times$ .

Therefore, for each $n$ we can construct a representation: $$\chi_{p,n} : G_{\Rats} \to \Gal(K_n/\Rats) \to (\Ints/p^n\Ints)^\times$$ where the first map is simply restriction to $K_n$ and the second map is an isomorphism. By the remarks above, the representations $\chi_{p,n}$ are coherent in a strong sense, i.e.

$$\chi_{p,n+1}(\sigma) \equiv \chi_{p,n}(\sigma) \mod p^n.$$

Therefore, one can construct a ``big'' Galois representation: $$\chi_p : G_{\Rats} \longrightarrow \Ints_p^\times$$ by requiring $\chi(\sigma) \equiv \chi_{p,n}(\sigma) \mod p^n$ , for every $n\geq 1$ .

One can rephrase the above definition as follows. Let $\sigma\in G_{\Rats}$ . We need to define a group homomorphism $\chi_p:G_{\Rats} \to \Ints_p^\times$ , so we need to first define $\chi_p(\sigma)$ and then check that it is a homomorphism. By the theory, $\sigma(\zeta_{p^n})$ is another primitive $p^n$ -th root of unity, thus $$\sigma(\zeta_{p^n})=\zeta_{p^n}^{t_n}$$ for some integer $1\leq t_n \leq p^n-1$ with $\gcd(t_n,p)=1$ (so $t_n$ is a unit modulo $p^n$ ). Moreover, $$\sigma(\zeta_{p^{n-1}})=\sigma(\zeta_{p^{n}}^p)=\zeta_{p^n}^{pt_n}=\zeta_{p^{n-1}}^{t_n}$$ Therefore, $t_n \equiv t_{n-1}$ modulo $p^{n-1}$ . Thus, we may define: $$\chi_p(\sigma) = \varprojlim t_n \in \Ints_p$$ and as we have shown, $\chi_p(\sigma)$ is a unit of $\Ints_p$ . Finally, the reader should check that $\chi_p$ is a group homomorphism.