CHAPTER I PLACES AND DIVISORS §1. FIELDS OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE Let K be a field. By a field of algebraic functions of one variable over K we mean a field R containing K as a subfield and which satisfies the following condition: R contains an element x which is transcendental over K, and R is algebraic of finite degree over K(x). The element x is of course not uniquely determined. If x' is any element of R which is transcendental over if, then R is algebraic of finite degree over K(x'). In fact, the degree of transcendency of R over K being one, R is algebraic over K(x'). In particular, x is algebraic over K(x')y and K(x, x') is of finite degree over K(x'). Since R is of finite degree over K{x), it is a fortiori of finite degree over K(xy a:'), which proves that it is of finite degree over K{x'). Those elements of R which are algebraic over /Care called constants. They form a certain subfield Kf of R, the field of constants. The field R is also a field of algebraic functions of one variable over K''. In fact, any element x of R which is transcendental over K is also transcendental over K\ and R, which is algebraic of finite degree over K(x), is also algebraic of finite degree over K'{x). It is important to keep in mind that, when discussing properties of a field R of algebraic functions of one variable, we shall consider in fact not properties of the field R alone but properties of the pair formed by K and R. For instance, let Z be any field, and set R = Z{x, y, z), where x and y are algebraically independent over Z and z is algebraic over Z{xy y). Set K\ = Z(x), K2 = Z(y). Then R is a field of algebraic functions of one variable over either one of the fields K\ or K2 but its properties as a field of algebraic functions of one variable over Ki may be quite different from its properties as a field of algebraic functions of one variable over Ki. However, when considering a field R of algebraic functions of one variable over a field K, the field of constants of R will appear more and more to be the essential object instead of K itself, which will gradually fade into the background. §2. PLACES Let R be a field and K a subfield of R. By a V-ring in R (over K) is meant a subring o of R which satisfies the following conditions: 1. o contains K 2. o is not identical with R\ 3. If a is an element of R not in o, then x~l is in o. Let o be a V-ring. Those elements in o which are not units in o (we call them "non-units") form an ideal p in o. In fact, if z is a non-unit and z e o, then xz 1 http://dx.doi.org/10.1090/surv/006/01
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