By Thomas W. Hungerford

Summary ALGEBRA: AN advent is meant for a primary undergraduate direction in sleek summary algebra. Its versatile layout makes it compatible for classes of varied lengths and diverse degrees of mathematical sophistication, starting from a standard summary algebra direction to at least one with a extra utilized style. The ebook is equipped round subject matters: mathematics and congruence. every one subject is built first for the integers, then for polynomials, and at last for jewelry and teams, so scholars can see the place many summary techniques come from, why they're vital, and the way they relate to 1 another.

New Features:

- A groups-first alternative that permits those that are looking to conceal teams sooner than earrings to take action easily.

- Proofs for novices within the early chapters, that are damaged into steps, every one of that is defined and proved in detail.

- within the middle path (chapters 1-8), there are 35% extra examples and thirteen% extra workouts.

**Read Online or Download Abstract Algebra: An Introduction PDF**

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**Additional resources for Abstract Algebra: An Introduction**

**Example text**

17. If pis prime and (a, 18. b) =p, then (a2, il) =? Prove or disprove each of the following statements: (a) If pis prime andpI (a2 + b2) andpI (c2 + Jl-), then p I (a2 - c2). (b) If pis prime andpI(a2 + b 2) andpI (c2 + t:P), then p I (a2 + c2). (c) If pis prime andpI aandpI (a2 + il), thenpI b. f B. 19. ' P'i [If, where p1, Pz, ... , Pk are distinct positive primes and each r1, s1 2: 0. Prove that aI b if and only if r1 s s1 for every i. · • · · · · p',; and b fl'i'P'i� 20. If a p�'P'iP? ] 21.

23. : 1. * 24. Let a, b, c Z. Prove that the equation E ax +by == c has integer solutions if (a, b) I c. and only if 25. (a) If a, b, u, v E Z are such that (b) Show by example that if au +bv = 1, prove that (a, b) = 1. au+ bv = d > 1, then (a, b) may not bed. 26. If a I c and b I c and (a, b) = d, prove that ab Ied. 27. If c lab and (c, a)= d, prove that cldb. 28. Prove that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. ] 29. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.

Hence m must have positive divi sors other than 1 or m, say m = ab with 1< a< m and 1< b< m. Since both a and bare less than m (the smallest element of S), neither a nor bis in S. By the definition of S, both a and bare the product of primes , say a= P1P2 · • ·p , and with r;;::; 1, s;;::; 1, and each p1, (/jprime. Therefore is a product of primes, so that m it S. We have reached a contradiction: m E S and m it S. Therefore, S must be empty. com). 10, which is proved on page 21, to do the factorization.