By Peter Fletcher

The authors imagine that scholars utilizing this article have already taken classes during which they constructed the ability of utilizing effects and arguments that others have conceived. this article selections up the place the others left off -- it develops the scholars skill to imagine mathematically and to tell apart mathematical considering from wishful pondering.

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To B' = I [Xl"'" Xn} by the correspondence defined in Def, 11 the conditions, Section 2, (6), (6'), (6), (3), (4), (5) remain valid though (e) need not. This transition maps the totality oj' elementl? of I [Xl' ••• ,xn ] upon the totality of the elements of [Xl"'" xnl ; it preserves equality and all relations arising from the rules of combinatwn, though not necessarily the I distinctness of the elements of I [x, ... , xn]. ;, I [Xl' ••. '1 of the above correspondence. 48 1. I. ) naturally follows from the fact that the correspondence rule of Def.

33 34 1. 1. Rings, Fields, Integral Domains 1\ow, let K be another field as specified in the theorem, a. an element of K and a b an arbitrary representation of 0. as a quotient of elements in I. Set up a correspondence whereby 0. is mapped on that element a of K which is represented by the same qnotient. By the above remarks and (1) this is it one-toone correspondence [Section 2, (5), (0'), (E), (Il')] between the totality of elements of K and K. By (2) and (3) it satisfies the conditions Section 2, (3) and (4).

The unique representations of these elements ill the form of Theorem 11 are called their 1UJ1'IIWl representatitJ1ls and the elements ([/'1"'" h n of I appearing thel'ein, the cQejJ'ideut8 of this representation. 46 1. I. Rings, Fields, Integral Domains In designating the x, as indeterminates we mean that each individual x" with respect to I, admits of no determination other ex; than the negative one, that no equation a,. + ,. =0 = 0 (with only finitely many coefficients 0) exists except the trivial one, where all al.