Book Chapter10.1016/S0049-237X(08)71549-0
Algebraic Function Fields and Non-Standard Arithmetic
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TL;DR: In this article, the authors discuss algebraic function fields and nonstandard arithmetic, and present a nonstandard model of the field of rational numbers Q for a higher order language, where * Q may be an ultra power of Q and an element of * Q which is not contained in Q.
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Abstract: Publisher Summary This chapter discusses the algebraic function fields and nonstandard arithmetic. Let * Q be a nonstandard model of the field of rational numbers Q , for a higher order language. In particular, * Q may be an ultra power of Q . Let (a) be an element of * Q which is not contained in Q . Then it is easy to verify that (a) is transcendental over Q . Thus, the field A = Q (a) c * Q is the field of rational functions with rational coefficients. An internal valuation of * Q is given either by a nonstandard prime number in * Q or by a standard prime number in Q (and * Q ) or by the Archimedean valuation of * Q .
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References
•Book
Introduction to the theory of algebraic functions of one variable
Claude Chevalley
- 01 Jan 1951
TL;DR: In this paper, the authors present an approach to algebraic geometry of curves treated as the theory of algebraic functions on the curve, which allows the author to consider curves over an arbitrary ground field.