By Thomas W. Hungerford

Summary ALGEBRA: AN creation is meant for a primary undergraduate path in smooth summary algebra. Its versatile layout makes it appropriate for classes of varied lengths and diversified degrees of mathematical sophistication, starting from a standard summary algebra path to at least one with a extra utilized style. The e-book is equipped round topics: mathematics and congruence. every one subject matter is constructed first for the integers, then for polynomials, and eventually for jewelry and teams, so scholars can see the place many summary recommendations come from, why they're very important, and the way they relate to at least one another.

New Features:

- A groups-first choice that permits those that are looking to hide teams sooner than jewelry to take action easily.

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

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

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

**Example text**

Mq ... _ .. Doo .. __�·>·--... _ .... ,_... ,,,__ ... ,. _ ... -tlajlll-. ll 26 Chapter 2 Congruence in Zand Modular Arithmetic The symbol used to denote congruence looks very much like an equal sign. This is no accident since the relation of congruence has many of the same properties as the relation of equality. For example, we know that equality is reflexive: a = a for every integer a; symmetric: if a=b, then b =a; transitive: if a= b and b = c, then a= c. We now see that congruence modulo n is also reflexive, symmetric, and transitive.

F4. nontrivial factor in common with 3 and another in common with 4. be has no common factors (except ±1) with either b or c, then there is a useful answer to the question. l. ar�io ..... to� .......... _ ....... -. lllWtrktkJas ... it. 4 If albc and {a, b) =1, then a le. 2 shows that au+ bv=1 for some integers u and v. Multiplying this equation by c shows that acu+ bcv= a jbc, so that be= ar for some r. Therefore c=acu+ bcv = acu + (ar)v=a(cu+ c. But ) ro . The first and last parts of this equation show that a I c.

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).