1.7. MODULAR ARITHMETIC 43 What are the objects Z n actually good for? First, they are devices for studying the integers. Congruence modulo n is a fundamental relation in the integers, and any statement concerning congruence modulo n is equiv-alent to a statement about Z n . Sometimes it is easier, or it provides better insight, to study a statement about congruence in the integers in terms of Z n . Second, the objects Z n are fundamental building blocks in several gen-eral algebraic theories, as we shall see later. For example, all ﬁnite ﬁelds are constructed using Z p for some prime p . Third, although the algebraic systems Z n were ﬁrst studied in the nine-teenth century without any view toward practical applications, simply be-cause they had a natural and necessary role to play in the development of algebra, they are now absolutely fundamental in modern digital engi-neering. Algorithms for digital communication, for error detection and
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.