By Thomas W. Hungerford

Summary ALGEBRA: AN creation is meant for a primary undergraduate path in glossy summary algebra. Its versatile layout makes it appropriate for classes of assorted lengths and varied degrees of mathematical sophistication, starting from a conventional summary algebra direction to at least one with a extra utilized style. The publication is geared up round subject matters: mathematics and congruence. each one subject matter is built first for the integers, then for polynomials, and eventually for jewelry and teams, so scholars can see the place many summary techniques come from, why they're very important, and the way they relate to 1 another.

New Features:

- A groups-first choice that allows those that are looking to conceal teams earlier than earrings to take action easily.

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

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

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

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**Extra info for Abstract Algebra: An Introduction**

**Sample text**

Wmlllarls,_,. _ ....... a11N:t... � ...... -. 2 Modular Arithmetic 36 5. For each a E Z, the equation a+ 6. If x a, = 0 has a solution in Z. 7. a(_bc) =(ab)c. 8. a(_h+ c) =ab+ ac and 9. 10. [Closurefor multiplication] bEZ, then abEZ. [Associative multiplication] (a + b)c =ac + be. [ DistributiVe laws] ab =ha [Commutative multiplication] [Multiplicative identity] a 1 =a =1 a • 11. If · ab =0, then a = 0 or b = 0. By using the tables in the preceding example, you can verify that the first ten of these properties hold in Zs and Z6 and that Property 11 holds in Zs and fails in �·But using tables is not a very efficient method of proof (especially for verify ing associativity or distributivity).

Induction is discussed in Appendix C. W-t.. a-:it. 3 Primes and Unique Factorization 17 32. Let a and b be integers, not both 0, and let tbe a positive integer. Prove that tis the least oommon multiple of a and b if and only if t satisfies these conditions: (i) (ii) a ] t and b I t; If a [ c and b Jc, then t I c. C. 33. ) 34. Prove that (a) (a, b)l(a + b, a - b); • (b) if a is odd and bis even, then (a, b) =(a+ b, a (c) if a and bare odd, then 2(a, b) = - b); (a + b, a - b). :l has at least four distinct divisors, namely 1, -1, n, -n.

EXAMPLE 3 The meaning of the symbol "[ ]" depends on the context. In congruence modulo 3, for instance, [2] {2 + 3klkEZ} = = { . . , -7, - 4, -1, 2, 5, 8, . . }, but in oongruence modulo 5 the congruence class [2] is the set {2 + 5k I kEZ} = { ... , -13, -8, -3, 2, 7, 12, . }. This ambiguity will not cause any difficulty when only one modulus is under discussion. On the few occasions when several moduli are discussed simultaneously, we avoid confusion by denoting the congruence class of modulo n by a [a]n· EXAMPLE 4 In congruence modulo 3, the congruence class [2] = {.