By Oscar Zariski

Zariski offers an excellent advent to this subject in algebra, including his personal insights.

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Then + ( r 2) (I) p(P) iS an integer, (2) P(P) >- 7/" ( P ) ()) P(T v) = ~- ( P ) -b 2 is v non-ne~ative oven p(~) = 2p(r) by the equation - 2. and if and only if 7" has no slngul=rlties. We are thus led to the following Def. s P ( z ) of Z is d e f i n e d by ~ot___e_e = (a) p(o) = l (b) p(K) = (Z 2) + l (c) p(-z)- I . Prop. z) + (z2) = 2p(z) - 2. z 2) - l. (b) p(Z) is an integer. , a cycle, then p(Z) ~ p(Z) = 7F(Z). prime Furthermore 7/" (Z) if and only if Z has no Bin~ular points. Z2)+(Z22)+2.

Ci where Then for all i,j. >'-~ > 0 which proves the o- (yjq/yiq), and for a suitable choice of the hence the poles of (2) Lq. , n. , n. 8: M is be the integral v such that Then ~ = ~ v~ Rv S c Rv aud s~_uce -58v( ~o ) -> 0 for all ~o yu~ ~ R' v N ~ for some u o. 00 = ~oYo q = ~iYi q. Iarge in it follows that Similarly ! Now RN+ q for large ! t ~ = ~h, Cot. 8: V and let write ~ t = hq. Therefore D~ L' which will prove q Then ~ CR' N+t' ~ ~ C R'tEN. is arit~uetically normal if and only if Let R ~ have t ~ s property, and, if This shows k[~'t] and hence 03 is integral over R.

D. k(W)/k = r - I. uniformizing coordinates of W. r - I Let {~i, "'" Then among the algebraically independent elements. are multiples of one irreducible polynomial denote by t. , ~ Hence be a prime divisorial the set is a principal ideal. Vw(t) = I. Let W I~I' "'" ~t be a simple s-dimensional subvariety of V, ~r ~ be a set of uniformizing coordinates of W and let such that -39{~ is a set of uniformizing parameters of W. , a we see that ~ ~ + a~ Then we . We have seen that ~ W k(W)-mcdule. By Prop.