Let $p>2$ be a prime, fix algebraic closures $\overline{\Rats}$ and $\overline{\Rats_p}$ , and fix an embedding of $\overline{\Rats}\hookrightarrow \overline{\Rats_p}$ . This embedding corresponds with an inclusion of the absolute Galois groups: $$\Gal(\overline{\Rats_p}/\Rats_p)\hookrightarrow \Gal(\overline{\Rats}/\Rats), \quad \sigma \mapsto \sigma|_{\overline{\Rats}}.$$ Let $I_p$ be the inertia subgroup of $\Gal(\overline{\Rats_p}/\Rats_p)$ which we regard as a subgroup of $\Gal(\overline{\Rats}/\Rats)$ via the injection above (for more information on the inertia subgroup at $p$ , $I_p$ , see the entry on Galois representations). Let $\F_{p^n}$ be the finite field of $p^n$ elements. The purpose of this entry is to define $\F_{p^n}$ -valued characters $\Psi_n$ , for every $n\geq 1$ : $$\Psi_n : I_p \longrightarrow \F_{p^n}^\times \cong \Ints/(p^n-1)\Ints$$ which we will refer to as the fundamental character of level $n$ of $I_p$ .

Next, we define the fundamental characters in more generality. Let $K_n/\Rats_p$ be the unique unramified field extension of degree $n$ (it is unique by local field theory). The residue field of $K_n$ is the field $k_n=\F_{p^n}$ (because $k$ must be an extension of degree $n$ of $\F_p$ ).

Proof. Clearly, the polynomial $x^{p^n-1}-1=0$ has $p^n-1$ distinct roots in $k_n=\F_{p^n}$ . Using Hensel's lemma, one can check that each root in $k_n$ lifts to an element of $K_n$ .

Let $K_n'=K_n((-p)^{\frac{1}{p^n-1}})$ . By the lemma, the $(p^n-1)$ th roots of unity are contained in $K_n$ . Therefore, the extension $K_n'/K_n$ is Galois. Moreover, by Kummer theory one has: $$\Gal(K_n'/K_n)=k_n^\times=\F_{p^n}^\times.$$ Notice that the fact that $K_n/\Rats_p$ is unramified implies that the inertia group $I_p$ injects into $\Gal(\overline{\Rats_p}/K_n)\hookrightarrow \Gal(\overline{\Rats_p}/\Rats_p)$ . Therefore there is a map: \begin{eqnarray} \label{psi} I_p \hookrightarrow \Gal(\overline{\Rats_p}/K_n)\to \Gal(K_n'/K_n) \to \F_{p^n}^\times \end{eqnarray}where the second map is simply given by restriction to $K_n'$ .

Definition 2   The fundamental character of level $n\geq 1$ is the map $\Psi_n : I_p \to \F_{p^n}^\times$ given by Eq. ().

Note from the author: I would like to thank Eknath Ghate for explaining this to me.