By Thomas W. Hungerford

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 assorted degrees of mathematical sophistication, starting from a conventional summary algebra direction to at least one with a extra utilized style. The booklet is geared up round subject matters: mathematics and congruence. each one subject is built first for the integers, then for polynomials, and at last for earrings and teams, so scholars can see the place many summary suggestions 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 prior to earrings to take action easily.

- Proofs for newcomers 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.

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**Example text**

X2 4. ls x4 =[lJ in Zs 5. x2 EB [3J 0 x®[2J = [OJ in Zt, 6. x2 EB [SJ 0 x = [OJin £9 7. x3 EB x2® x®[lJ =[OJ in Zs 8. x3 9. + x2 =[2J inZ10 (a) Find an element [aJ in Z7 such that every nonzero element of Z7 is a power of [aJ. (c) Can you do part (a) in�? s. 10. 7. 11. Solve the following equations. (a) x®x ®x =[OJinZ 3 (b) x®x ®x ®x =[OJ inZ4 (c) xEBx®x®x®x =[OJ in Zs 12. Prove or disprove: If [aJ 0 [bJ = [OJin Z,,, then [a] = [OJ or [b] = (O]. 13. Prove or disprove: If [a] 0 [bJ =[a] 0 [cJand [a] :f: [ O ] in Z,,, then [bJ =[c].

Let n be a positive integer. Prove that a and cleave the same remainder when n if and only if a c = nk for some integer k. divided by - 11. Prove the following version of the Division Algorithm, which holds for both positive and negative divisors. Extended Division Algorithm: Let a andb be integers with b :# 0. Then there exist unique integers q a11d rsuch that a= hq + randO s r < JbJ. ] • lb 1- Then consider two cases Divisibility An important case of division occurs when the remainder is 0, that is, when the divisor is a factor of the dividend.

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 ..