Cyclotomic character
= Cyclotomic character
{title2=$\kappa:G_F\to\mathbb Z_p^\times$}
The cyclotomic character is defined by $g(\zeta)=\zeta^{\kappa(g)}$ on compatible $p$-power <roots of unity>. On the full cyclotomic tower of $\mathbb Q$, it identifies the <Galois group> with $\mathbb Z_p^\times$. Its finite-order part is the <Teichmüller character> and its pro-$p$ part acts through $1+p\mathbb Z_p$, or $1+4\mathbb Z_2$.