POLYNOMIAL RING S AN D AFFIN E SPACE S 3 in one variable X over k is again a unique factorization domain, and consequently F is a unique factorization domain. COROLLARY 1.5 . Ifk is a field, then Fis a unique factorization domain. It is well known that : THEOREM 1.6 . Ifkis a noetherian ring, then F is also a noetherian ring. Since this is also written in many textbooks, we omit the proof. In connection with these two results stated above, we relate THEOREM 1.7 . Ifk is a noetherian normal domain, then F is also a noetherian normal domain. Here, by a normal domain, we understand a n integral domain which is integrally closed in its field o f fractions . In order to prove the theorem, besides Corollary 1. 5 an d Theorem 1.6 , we need THEOREM 1.8 . Ifk is a noetherian normal domain, then (i ) for every prime ideal P of height one (i. e., P is a prime ideal and there are no prime ideals properly between P and zero), the ring kp = {a/b \a, b E k, b £ P} is a discrete valuation ring and (ii) k is the intersection of all such k p . Since this is a part of a well-known characterization o f normality of a noetherian domai n and is written in many standard textbooks , we omit the proof. For the proof of Theorem 1.7 , we begin with Gauss ' lemma: IJ:MMA 1.9 . Ifkis a unique factorization domain, then k is a normal domain. PROOF. Assum e that f/g (f, g E k, g 0) is integral over k. W e may assume that /and #have no common prime factor. Ther e are cv ... , c d Ek suc h that (f/g)d + c 1 (f/g)d~l + " " + c d = 0 - The n /** + cj d ~ l g + + c d ^ = 0 and f2 i s divisible by any prime facto r p of g. The n / i s divisible b y p an d therefor e / an d g have a common facto r p, a contradic- tion. Thu s g has no prime factor , i . e., g is a unit. Q . E . D. PROOF OF THEOREM 1.7. For each prime ideal P of height one in k, we consider Fp = kp[Xv . . . , Xn]. Thi s is a unique factorizatio n domai n b y Corollary 1.5 , hence i s a normal domai n b y Lemm a 1.9 . Let/b e a n element o f th e fiel d o f fraction s o f F = k[Xlf . . . , Xn] suc h that / i s integral ove r F. Th e normality o f Fp implie s tha t / i s in F*, thu s / E ClpF* = F. Q . E. D. Now we discuss results closel y relate d t o th e famou s theore m know n a s the normaliza- tion theorem of Noether whic h is stated a s follows: THEOREM 1.10 . Assume that a ring R is generated by a finite number of elemen ts over a field k. Then there are elements z v . . . , zt of R such that (i ) zv . . . , zt are algebraically independent over k (i.e.,k[z lf ..., z t ] is naturally isomorphic to the polynomial ring in t variables over k) and (ii) R is integral over k[z v . . . , z t ]. If R = k[X x , . . . , Xn] and if P is a nonzero ideal in R, then we may choose z x from P.
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