By Ralph Stocker, Heiner Zieschang

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Y + x . z for all x, y, z in R. e. for any choices of a, b the relation ab = 0 implies that a = 0 or b = O. Cancellation is possible in domains, meaning that if ab = ac and a =1= 0 then b = c; just observe that ab = ac is equivalent to a(b - c) = O. e. an element u- 1 (necessarily unique) with u'u- 1 = u-1·u = 1. A commutative ring R is a field when every element a =1= 0 has a multiplicative inverse; thus a field is automatically a domain. Important examples of domains for us are Z, (Q, IR. and C, all with the usual definitions of addition and multiplication; of these (Q, IR.

A mapping that preserves both distance and orientation.

Eliminating s, t from (*) via Cramer's Rule we obtain the conic in the (x,y)-plane given by ((d - X)x - Yy)2 + (-Yx + Xy)2 = F 2. More precisely, we have a family of conics, depending on the choice of tracing point P = (X, Y). The nature of the conic depends on this choice. For instance, the conic is an ellipse if and only if P does not lie on the circle F = O. ) However, when P does lie on the circle F = 0 the equation degenerates to a perfect square (Y x - Xy)2 = 0, representing a 'repeated line', in fact the line through P and the origin.

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