- an algebraic number field, i.e., a finite extension of Q, or
- a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq(T), the field of rational functions in one variable over the finite field with q elements.
Analogies between the two classes of fields
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 field of fractions 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:
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 Dedekind and Heinrich M. Weber in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for curves over finite fields settled by André Weil in 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 theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example. The analogy was also influential in the development of Iwasawa theory and the Main Conjecture. The proof of the fundamental lemma in the Langlands program also made use of techniques that reduced the number field case to the function field case.
- Artin, Emil; Whaples, George (1945), "Axiomatic characterization of fields by the product formula for valuations", Bull. Amer. Math. Soc. 51: 469–492, doi:10.1090/S0002-9904-1945-08383-9, MR 0013145
- Artin, Emil; Whaples, George (1946), "A note on axiomatic characterization of fields", Bull. Amer. Math. Soc. 52: 245–247, doi:10.1090/S0002-9904-1946-08549-3, MR 0015382
- J.W.S. Cassels, "Global fields", in J.W.S. Cassels and A. Frohlich (eds), 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.
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, Zbl 1136.11001, MR 2392026