Home > Global field
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 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:
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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 hypothesisThe Riemann hypothesis first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ s . It is one of the most important open problems of contemporary mathematics; a $1,000,000 prize ha is known to be true (by work of André WeilAndre Weil ( May 6, 1906 August 6, 1998) was one of the great mathematicians of the 20th century, a founding member of the influential Bourbaki group. He was brother of the philosopher Simone Weil. Born in Paris, he studied in Paris, Rome and Gottingen an and others), and there is great interest in developing parallel techniques for number fields.
Field theoryField theory (mathematics), the theory of the algebraic concept of field. Field theory (physics), a physical theory which employs fields in the physical sense. Algebraic number theory
