By Aichinger E.

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By the repeated use of this process, the x11De for some c becomes a separating element of K. D. Hence, from the above proof we obtain the following corollary. §2. 1. If the coefficient field k is a perfect field of characteristic p # 0 and if x is not a separating element of K, then x1',, the pth root of x, is contained in K. For the case of a perfect field k we may take a separating element x so that K = k(x, y) is as described earlier. Then one may take the separating element y11pa of K so that K = k(x, yl/pe) , instead of y.

This is because when x is transcendental over k(x'), then k(x, x') is the field of rational functions of two variables with coefficients in k. Hence, x' also has to be a transcendental element over k(x), which contradicts condition (i). Therefore x is algebraic over k(x'),and [k(x, x') ; k(x')] is finite. From condition (i), [K: k(x, x')] is finite, hence K is also finite over k(x'). 1. Thus, it is desirable to choose an element x so that the extension K over k(x) may have properties as simple as possible to facilitate the study 33 2.

N. 5). If n C = C'X 1' ... xnn , 1. PREPARATION FROM VALUATION THEORY 8 then c' E o. Let a 1 be the product of c' and x1 f for e, > 0, and let a2 be the product of x1 ' for e1 <0. Now we have c= a1 a2 , a1,a2Eo. This proves the second half of (i). D. 2. 3. Suppose o is a subring of K such that (i) an arbitrary element c of K can be written as a quotient of elements of o, and (ii) any o-ideal of K can be decomposed uniquely as a product of prime ideals, and furthermore if J c J', then we have J = J'J" for some ideal J"Co.

### 2-affine complete algebras need not be affine complete by Aichinger E.

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