By O. Zariski, P. Samuel
Covers issues comparable to valuation conception; conception of polynomial and gear sequence jewelry; and native algebra. This quantity contains the algebro-geometric connections and purposes of the in simple terms algebraic fabric.
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Extra resources for Commutative Algebra II
Valuation of K. By the rank of v we mean the rank of any place such VALUATION THEORY Ch. VI (see § 3, Definition 1). We proceed to interpret the rank that Kai = of v directly in terms of the value group P of v. A non-empty subset of P is called a segment if it has the following property: if an element a of P belongs to zl, then all the elements fi of P which lie between a and — a (the element — a included) also belong to A subset of 1' is called an isolated subgroup of P if is a segment and a proper subgroup of P.
That every elen'ient x of x 0, can be put (uniquely) in the form atlz, 0 and a is a unit. This shows that the principal ideals (ta), where n Hence the maximal ideal all the proper ideals of , are n = 1, 2, . . whence v is of rank 1. is the only proper prime ideal of (t) of Furthermore, it is immediately seen that if K' denotes, as usual, the then multiplicative group of the field K and E is the set of units in the quotient group K'/E, written additively, is isomorphic to the group of integers.
Now, by assumption, there exists a place having center q and such that oo. For such we will have a place 0 since afi =0, i =0, 1,.. , j =1, and 0 (in view of the assumption made on the coefficients a0, a1,... ). Let be and also Therefore the element of o does not belong to q, and consequently This completes the proof of the lemma. t= — E 0q. We note the following consequence of the lemma: COROLLARY. Let o be an integrally closed integral domain, let K be the quotient field of o and let q be a prime ideal in o.