By Thomas W. Hungerford
Summary ALGEBRA: AN advent is meant for a primary undergraduate direction in glossy summary algebra. Its versatile layout makes it compatible for classes of varied lengths and various degrees of mathematical sophistication, starting from a standard summary algebra direction to 1 with a extra utilized taste. The booklet is equipped round topics: mathematics and congruence. every one topic is constructed first for the integers, then for polynomials, and eventually for jewelry and teams, so scholars can see the place many summary options come from, why they're very important, and the way they relate to at least one another.
- A groups-first alternative that allows those that are looking to disguise teams earlier than jewelry to take action easily.
- Proofs for newbies within the early chapters, that are damaged into steps, every one of that's 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 resources for Abstract Algebra: An Introduction
P,) = q3q-4 • • • q8. 6 and elimi nating one prime on each side at every step. If r = s, then this process p1 = ±q" P2 = ±q2, • • • , p, = ±q,. So to complete the proof of the theorem, we must show that r = s. The leads to the desired conclusion: proof that r = s is a proof by contradiction: We assume that r =fo s (which means that r > s or that r < s), and show that this assumption leads to a contradiction. First, suppose that r > s. Then afters steps of the preceding process, all the q's will have been eliminated and the equation will read ±Pi+tP1+2 · · ·P, = L This equ ation says (among other things) that p, 11.
2 Divis I bl lity 15 (b) Every common divisor of b and r is also a common divisor of a and b. (c) (a, b) = (b, r). 14. Find the smallest positive integer in the given set. ] E Z} 15. The Euclidean Algorithm is an efficient way to find (a, b) for any positive integers a and b. It only requires you to apply the Division Algorithm several times until you reach the gcd, as illustrated here for (524, 148). (a) Verify that the following statements are correct. :,. ] nonzero 4·2 + 0 (b) Use part (a) and Exercises 13 and Example 4 to prove that (524, 148) = (148, 80) = (80, 68) = (68, 12)=(12, 8)=(8, 4)=(4, 0)=4.
Ootbll� �-w :la11'fdiiwia:r-t. _:ligl:U� it. .. 3 7. 8. Primes and Unique Factorization If a, b, c are integers and p is a prime that divides both a and thatplbor ple. + a 23 be, prove (a) Verify that x - l is a factor of X' - 1. (b) If n is a positive integer , prove that the prime factorization of 2'bi 311 - 1 includes 11as one of the prime factors. J · • • 9. Letpbe an integer other than 0, ± l. Prove that p is prime if and only if it has this property: Whenever r and s are integers such that p = rs, then r = ±1 ors= ±1.
Abstract Algebra: An Introduction by Thomas W. Hungerford