Global field - Definition and Overview

The term global field refers to either of the following:

• a number field, i.e., a finite extension of Q or
• the function field of an algebraic curve over a finite field, i.e., a finitely generated field of characteristic p>0 of transcendence degree 1.

There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the quotient field of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the product formula for non-zero elements x:

<math>\prod_v |x|_v = 1.<math>

The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. In particular, it is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example. The function field analogue of the Riemann hypothesis is known to be true (by work of André Weil and others), and there is great interest in developing parallel techniques for number fields.