Exploring Mathematics by JOHN MEIER & DEREK SMITH

Author:JOHN MEIER & DEREK SMITH
Language: eng
Format: epub
Publisher: Cambridge University Press

But then , so is a multiplicative inverse of in .

() If has a multiplicative inverse, then there exists an such that . But this means that there is some such that , and therefore . Thus .

DEFINITION 6.24. Define to be the subset of consisting of equivalence classes whose representatives are relatively prime to m:

In the study of arithmetic systems, elements that have multiplicative inverses are called units. The notation comes from the fact that we are looking at nits in .

Exercise 6.30 Show that if , then has exactly one multiplicative inverse.

The set of units in is , where we are abusing notation by dropping the brackets. The fact that all of these elements are units is verified by the multiplication table for shown in Figure 4, as every row contains a 1.

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