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

where $\mathbb{Z}_{p}^{\times}$ is the group of units of $\mathbb{Z}_{p}$, the $p$-adic integers. $\chi_{p}$ is a $\mathbb{Z}_{p}^{\times}$ valued character, usually called the cyclotomic character of $G_{{\mathbb{Q}}}$, or the $p$-adic cyclotomic Galois representation of $G_{{\mathbb{Q}}}$. Here is the construction:

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

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

Therefore, for each $n$ we can construct a representation:

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.

Therefore, one can construct a “big” Galois representation:

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 $\sigma\in G_{{\mathbb{Q}}}$. We need to define a group homomorphism $\chi_{p}:G_{{\mathbb{Q}}}\to\mathbb{Z}_{p}^{\times}$, 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

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,

Therefore, $t_{n}\equiv t_{{n-1}}$ modulo $p^{{n-1}}$. Thus, we may define:

and as we have shown, $\chi_{p}(\sigma)$ is a unit of $\mathbb{Z}_{p}$. Finally, the reader should check that $\chi_{p}$ is a group homomorphism.