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By Helmut Hasse

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D) Section 2, (e') is not satisfied. for instance, if we choose for 1 the field K (Section 1, Example ,1,) consisting only of 0 and e. For, in this case to the two distinct elements x x 2 and 0 of K [x] there corresponds the same function 0 in the sense of X2 is also zero for all x in K (that is, analysis, because x for x=O and x=e). + + Moreover, in Vol. 2, Theorem 49, [208] and Vol. 3. Section 4, Exer. 7, 8 as well as Section 1, Exer. 9 we will see. that Section 2, (8') is satisfied if and only if I has infinitely many elements.

XII) = 9(Xl , . . , XII) as function value for the system of elements :r l' . • , x" over I. Therefore, we make the further convention that the symbol == (contrariwise, =1=) shall be used to designate the first of the two statements. 12 On the basis of what has already been agreed upon, however, we can and will use the expression f=9 as equivalent to f(x l , ••• ,xlI )=g(x l , •• "x lI ) . By these conventions it unequivocally follows from the chosen designation which of the two possible interpretations of Xl"'" X I1 is valid in a relation of equality or inequality.

It was stated at the end of the introduction that the deduction of such statements was the principal problem of modern algebra. This exposition may seem to imply that in algebra no distinction whatever is drawn between isomorphic domains. This implication has =, 32 1. I. Rings, FLelds, Integral Domams to be qualified. True, the basic problem of algebra, as formulated in the introduction, is indifferent to whether it deals with a domain B or a B' isomorphic to B. However, if the isomorphic domains Band B' are both subdomains of another domain B , then a distinction between them arises in a natural way, namely, their elements are distinct (with respect to considerations inside B ) in virtue of the individuality of the elements of B (cf.

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