Roots of Unity in Integral Domains

A commutative associative ring D with 1 is called an integral domain if the product of any two elements in D 0 is in D 0.  

Throughout this section, the letter D stands for an integral domain. 

our claim follows from the hypotheses that d — 1 0 and dn — 1=0. 

We are assuming that d2n = 1. Thus, we obtain from Lemma 8.7.1 that 

Lemma 8.7.4 Assume that D has characteristic 0, and let us denote by Z the smallest unitary subring of D. Let d be an element in D for which there exists a positive integer n with dA = 1. Then, ifd+d-1 G Z, d4 = 1 or d6 = 1. 

as we are assuming that d + d G Z, induction yields that, for each integer z, d + d G E. (Note that dP + dr° G Z.) 

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In the last section of this chapter, in Section 8.7, we present identities about roots of unity in integral domains. The results will be useful in Section 12.4 where we shall investigate Coxeter sets of cardinality 2. 

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