By Thomas W. Hungerford

ISBN-10: 1111569622

ISBN-13: 9781111569624

Summary ALGEBRA: AN creation is meant for a primary undergraduate path in glossy summary algebra. Its versatile layout makes it compatible for classes of assorted lengths and diverse degrees of mathematical sophistication, starting from a standard summary algebra path to at least one with a extra utilized style. The e-book is geared up round subject matters: 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 suggestions come from, why they're vital, and the way they relate to at least one another.

New Features:

- A groups-first alternative that allows those that are looking to disguise teams prior to jewelry to take action easily.

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

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

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

**Example text**

A) a= 8,12 6,493; b= 541 (b) a= -9,217,645; b= 617 (c) a= 171,819,920;b = 4321 5. Let a be any integer and let b and e be positive integers.. Suppose that when a is divided by b, the quotient is q and the remainder is r, so that a = bq + r and 0 s r < b. If ae is divided by be, show that the quotient is q and the remainder is re. B. 6. Let a, b, e, and q be as in Exercise 5. Suppose that when q is divided by e, the quotient is k. Prove that when a is divided by be, then the quotient is also k.

So r > s cannot occur. A similar argument = shows that the assumption r < s also leads to a contraction and, hence, cannot occur. Therefore, r = s is the only possibility, and the theorem is proved. com) factors an integer quickly. 94017 TI calculators on our website as a product of primes relatively n · 7 · 112 • 37, as shown in F igure 1. 3 7 u Done FIGURE1 On Maple, the command ifactor(n ); will produce the prime factorization of n. 9 Every integer n = p1p2p3 Pa :S • • • • • n > 1 can be written in one and only one way in the form ·Pr• where the p1 are p ositive primes such that p1 s p2 s S Pr· Proof ..

Then [a + c] Proof"' Since [a] [c] = = and [b + d] [ac] = [bd]. 3. Similarly, [d] implies that c = d (mod n). 2, = a + c = b + d (mod n) Hence, and ac= bd(modn). 3 again, [a + c] = [b + d] and [ac] = [bd]. 6, we know that the following formal definition of addition and multiplication of classes is independent of the choice of representatives from each class: Definition Addition and multiplication in Zn are defined by [a] EB [c] = [a+ c] and [a] 0 [c) = [ac]. l. wtdil«blJll"I. 10� ...... tmn... &lbmbll......

### Abstract Algebra: An Introduction by Thomas W. Hungerford

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