Adéle

An element of an Adéle Group, sometimes called a Repartition in older literature. Adéles arise in both Number Fields and Function Fields. The adéles of a Number Field are the additive Subgroups of all elements in $\prod k_\nu$, where $\nu$ is the Place, whose Absolute Value is $<1$ at all but finitely many $\nu$s.

Let $F$ be a Function Field of algebraic functions of one variable. Then a Map $r$ which assigns to every Place $P$ of $F$ an element $r(P)$ of $F$ such that there are only a finite number of Places $P$ for which $\nu_P(r(P))<0$.

See also Idele

References

Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc., p. 25, 1951.

Knapp, A. W. ``Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.