Place of a global field

Definition

Let $F$ be a global field. A place of $F$ is an equivalence class of nontrivial absolute values on $F$ , where two absolute values are equivalent when they induce the same topology on $F$ .

Choosing a representative $|\cdot|_v$ gives a metric and hence the completion $F_v$ ; equivalent representatives give canonically isomorphic topological fields.

Archimedean and nonarchimedean places

The archimedean places occur only for number fields and come from real or complex embeddings. Every other place is represented by a nonarchimedean absolute value, equivalently by a discrete valuation after normalization.

For a global function field, the places correspond to closed points of its smooth projective curve.

Normalization

An equivalence class does not choose a numerical scale. In global arithmetic one normally selects representatives satisfying the product formula. Local formulas involving $q_v$ , Frobenius, or Haar measure therefore state their normalization separately.

References

Jürgen Neukirch, Algebraic Number Theory, Springer, 1999, Chapter II, §§4--5.
André Weil, Basic Number Theory, third edition, Springer, 1974, Chapter I.