By Thomas A. Whitelaw B.Sc., Ph.D. (auth.)

ISBN-10: 0216922593

ISBN-13: 9780216922594

ISBN-10: 1461572843

ISBN-13: 9781461572848

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Lber Cx of 2 m (x E 2). Because Cx "# Co, x =1= 0 (mod m) and so mtx. 3) there are in te gers u, u such that ux + um = 1. e. e. CuC x = CxC u = CI (= the unity of 2 m). This shows that Cu is an inverse of Cx in 2 m. ) is a field in this case. It is a useful easy exercise to make up addition and multiplication tables for ·2mfor selected values of m. As an illustration the addition and multiplication tables for 2 5 are given below. e. g. Fermat's theorem on congruences, which states that if pis a prime and n is an integer not divisible by p, then nP-1 == 1 (mod p).

Qs; and also that PI ~ ql' We note that PI #- ql; for, if PI were equal to ql' we would have m - PI = P2P3···Pr = q2q3···qs giving two genuinely different prime product expressions for the integer m/pI' which, however, is less than the least counterexample m. Hence PI < ql' and so, since ql ~ q2 ~ ... ~ qs' PI is less than qj for eachj. 6, PI must divide at least one of ql' qz, ... , qs' We thus have a contradiction, and the stated result folIows. 14. ", we find ourselves wanting to ignore integral multiples of 12.

Obtain a contradiction by showing that no prime divides the integer PI P2 ... J Prove by a similar approach that there are infinitely many primes that are congruent to 3 (mod 4). CHAPTER THREE EQUIVALENCE RELATIONS ANDEQUIV ALENCE CLASSES 15. Relations in general Underlying the discussion which we now begin is the widespread occurrence in mathematics of sentences of the form x is related [in some particular way] to y in which x, y denote elements of some given set S. Examples in the case S include: x=y; x

### Introduction to Abstract Algebra by Thomas A. Whitelaw B.Sc., Ph.D. (auth.)

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