- Global field
mathematics, the term global field refers to either of the following:
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 compactfields (see local fields). Every field of either type can be realized as the field of fractionsof a Dedekind domainin which every non-zero ideal is of finite index. In each case, one has the "product formula" for non-zero elements "x"::
The analogy between the two kinds of fields has been a strong motivating force in
algebraic number theory. The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekindand Heinrich M. Weberin the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curveis mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for local zeta-functionssettled by André Weilin 1940. The terminology may be due to Weil, who wrote his "Basic Number Theory" (1967) in part to work out the parallelism.
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 theoryand its exploitation by Gerd Faltingsin his proof of the Mordell conjectureis a dramatic example.
J.W.S. Cassels, "Global fields", in J.W.S. Cassels and A. Frohlich(edd), "Algebraic number theory", Academic Press, 1973. Chap.II, pp.45-84.
* J.W.S. Cassels, "Local fields",
Cambridge University Press, 1986, ISBN 0-521-31525-5. P.56.
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