By Thomas W. Hungerford
Summary ALGEBRA: AN advent is meant for a primary undergraduate path in glossy summary algebra. Its versatile layout makes it appropriate for classes of assorted lengths and diverse degrees of mathematical sophistication, starting from a conventional summary algebra direction to 1 with a extra utilized style. The ebook is geared up round subject matters: mathematics and congruence. each one topic is built first for the integers, then for polynomials, and eventually for earrings and teams, so scholars can see the place many summary innovations come from, why they're vital, and the way they relate to 1 another.
- A groups-first choice that allows those that are looking to disguise teams prior to earrings to take action easily.
- Proofs for rookies 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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Additional resources for Abstract Algebra: An Introduction
Prove that n is a perfect square if and only if each r1 is even. · · · 23. Prove that aI b if and only if a2 i b2. ] .... �J*I.. -. lldladlll....... -. 24 Chapter 1 Arithmetic in l. Revisited 24. Prove that a I b if and only if d' I b". 25. Let p be prime and 1 s k
Dl&d... ,-tlmlJlif�:dgbll�----k 30 Chapter 2 Definition Congruence in 1. and Modular Arithmetic The set of all congruence classes modulo n is denoted Zn (which is read "Z mod n"). There are several points to be careful about here. The elements of Zh are classes, not single integers. So the statement  EZh is true, but the statement 5 E Zti is not. F urthermore, every element of Zti can be denoted in many different ways. 3,  of = (5] == (mod 2 = 14 (mod 3). 5, which says in effect that The set Z� has exactly n elements.
11. If a, bare integers such that that a= a= b(modp) for e very positive prime p, prove b. ] 12. : = 13. Prove that =  in �· a = b (mod n) if and only if a and b leave the same remainder when divided by n. 14. (a) Prove or disprove: If ab= 0 (mod n), then a= 0 (mod n) or b= 0 (mod n). (b) Do part (a) when n is prime. 15. If (a, n) 16. If [a ] = = 1, prove that there is an integer bsuch that ab = 1 (mod n). , prove that (a, n) = 1. Show by example that the converse may be false. 17. Prove that 10" = (-lf (mod 11) for every positive n.