By Thomas W. Hungerford

ISBN-10: 1111569622

ISBN-13: 9781111569624

Summary ALGEBRA: AN creation is meant for a primary undergraduate direction in glossy summary algebra. Its versatile layout makes it appropriate for classes of assorted lengths and diverse degrees of mathematical sophistication, starting from a standard summary algebra direction to at least one with a extra utilized style. The e-book is prepared round issues: mathematics and congruence. each one subject matter is constructed first for the integers, then for polynomials, and at last for jewelry and teams, so scholars can see the place many summary suggestions come from, why they're vital, and the way they relate to 1 another.
New Features:
- A groups-first choice that allows those that are looking to disguise teams prior to 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 routines.

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Example text

Justify your answer. 10. If (a, 0)= 1, what can a possibly be? 11. If n E Z, what are the possible values of (a) (n, n + 2) (b) (n,n + 6) 12. Suppose that (a, b)= 1and(a, c) = 1. Are any of the following statements false? Justify your answers. (a) (ab, a) =1 (b) (b, c) =1 (c) (ab, c) =1 13. Suppose that a, b, q, and r are integers such that a= bq+ r. Prove each of the following statements. (a) Every common divisor c of a and bis also a common divisor of b and r. [Hint: For some integers sand t, we have a=cs and b=ct.

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). So the proof that Properties 1-10 hold for any Z,.

What numbers are equal to a given number a-just a itself. With congr uence, however, the story is different and leads to some interesting consequences. Definition Let a and (denoted n be integers with n > 0. The congruence class of a modulo n [a]) is the set of all those integers that are congruent to a modulo n, thatlsi [a]= {bJbEZ and b s a (mod n)}. To say that b that b =a (mod n) means that b-a = kn = a+ kn. Z} = {a +kn I kEl}. *The first two lines of this proof for a suitable expression are a standard algebraic technique: Rewrite 0 in the form -X + X X.

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Abstract Algebra: An Introduction by Thomas W. Hungerford


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