Root number of a local field (reviewed)

For a finite extension $K/\mathbb{Q}_p$, we give the local root number, or local sign, $\epsilon(K) \in \{1,i,-1,-i\}$.

We start with a general description where we allow $p=\infty$, writing $\Q_\infty = \R$. Local root numbers, $\epsilon(\rho) \in \C^\times$, are defined for representations $\rho : \mathrm{Gal}(\overline{\Q}_p/\Q_p)\rightarrow \GL_n(\C)$. They have absolute value $1$, are multiplicative in the sense that $\epsilon(\rho_1 \oplus \rho_2) = \epsilon(\rho_1) \epsilon(\rho_2)$, and for $p \neq \infty$, any unramified representation has root number $1$. The root number for the trivial and sign characters of $\mathrm{Gal}(\C/\R)$ are $1$ and $-i$ respectively. Finally, if $\rho : \mathrm{Gal}(\overline{\Q}/\Q) \rightarrow \GL_n(\C)$ is a global representation whose restrictions to decomposition groups are denoted by $\rho_p$, the global root number $\epsilon(\rho)$ equals the product of local root numbers $\prod \epsilon(\rho_p)$, and $\epsilon(\rho)$ figures into the functional equation of the Artin $L$-function $L(\rho,s)$.

If $K$ is an $n$-dimensional $p$-adic algebra then $\mathrm{Gal}(\overline{\Q}_p/\Q_p)$ acts on the set of its $n$ embeddings into $\overline{\Q}_p$. One thus has a representation $\rho_K : \mathrm{Gal}(\overline{\Q}_p/\Q_p) \rightarrow S_n \subset \GL_n(\C)$. For a local field $K$, we give the corresponding root number $\epsilon(K) := \epsilon(\rho_K)$. These particular root numbers play a central role in Galois embedding problems. Finally, we note that for a global number field $K$, the global root number $\epsilon(K) = \prod \epsilon(K_p)$ is always $1$.