Place of a number field

A place of a number field $K$ is an equivalence class of absolute values on a field. Its Archimedean places come from real embeddings and conjugate pairs of complex embeddings. Its non-Archimedean places correspond to nonzero prime ideals of the ring of integers of a number field.

Equivalence is taken among absolute values on a field, so changing a representative within one place does not change the associated topology.

Such a field embedding pulls the p-adic absolute value back to a finite place of a number field above $p$. It extends to an embedding of the corresponding completion, whose image is a finite extension of the P-adic numbers.

Finite places of a number field above $p$ correspond to absolute Galois group orbits of embeddings of a number field into a p-adic algebraic closure. The absolute value is invariant under the group. Conversely, embeddings inducing the same place extend to embeddings of the same local completion and are conjugate.

A real embedding $K\hookrightarrow \mathbb{R}$ gives a real Archimedean place of local degree $1$. A conjugate pair of complex embeddings $K\hookrightarrow \mathbb{C}$ gives a complex Archimedean place of local degree $2$.

A nonzero element of a number field is totally positive when its image is positive under every real field embedding. In a totally real number field, the principal fractional ideals having a totally positive generator form the subgroup used to define the narrow ideal class group. Signs of different generators are related by the unit signature map.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index of a prime ideal $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

For a place $v$ of a number field $K$, the local degree is $d_v=[K_v:{\mathbb{Q}}_v]$. Thus $d_v=1$ at a real place, $d_v=2$ at a complex place, and $d_v=e_v{f}_v$ at a finite place over $p$. For each rational place, the local degrees above it sum to $[K:\mathbb{Q}]$.