Global field - Knowlpedia

A global field is a field isomorphic either to a number field or to a global function field. Equivalently, it is a finite extension of $\mathbb Q$ or of $\mathbb F_q(t)$ for some prime power $q$ .

Places and completions

Every nontrivial place $v$ of a global field $F$ determines a completion $F_v$ . These completions are local fields. Thus the arithmetic of one global field is studied simultaneously through all of its localizations.

The two cases

For a number field, the archimedean completions are $\mathbb R$ or $\mathbb C$ , and its nonarchimedean completions are finite extensions of $\mathbb Q_p$ . A global function field has no archimedean places; all of its completions have positive characteristic.

Langlands role

Global automorphic objects live over $F$ or its adele ring, while their local components live over the fields $F_v$ . The number-field and function-field versions of the Langlands program share this local--global architecture even though their geometric tools differ.

References

John W. S. Cassels and Albrecht Fröhlich, eds., Algebraic Number Theory, Academic Press, 1967.
Michael Rosen, Number Theory in Function Fields, Springer, 2002.