By Martyn Dixon, Leonid Kurdachenko, Igor Subbotin

ISBN-10: 0470640537

ISBN-13: 9780470640531

Publish 12 months note: First released September twenty seventh 2010

Explore the most algebraic constructions and quantity platforms that play a valuable function around the box of mathematics

Algebra and quantity concept are strong branches of contemporary arithmetic on the vanguard of present mathematical examine, and every performs an more and more major function in numerous branches of arithmetic, from geometry and topology to computing and communications. in accordance with the authors' wide event in the box, Algebra and quantity Theory has an leading edge procedure that integrates 3 disciplines--linear algebra, summary algebra, and quantity theory--into one complete and fluid presentation, facilitating a deeper realizing of the subject and bettering readers' retention of the most concepts.

The publication starts off with an creation to the weather of set concept. subsequent, the authors speak about matrices, determinants, and parts of box thought, together with initial info on the topic of integers and intricate numbers. next chapters discover key principles when it comes to linear algebra resembling vector areas, linear mapping, and bilinear varieties. The publication explores the advance of the most principles of algebraic constructions and concludes with purposes of algebraic principles to quantity theory.

Interesting functions are supplied all through to illustrate the relevance of the mentioned strategies. additionally, bankruptcy routines permit readers to check their comprehension of the awarded material.

Algebra and quantity Theory is a superb ebook for classes on linear algebra, summary algebra, and quantity concept on the upper-undergraduate point. it's also a worthy reference for researchers operating in several fields of arithmetic, laptop technology, and engineering in addition to for people getting ready for a profession in arithmetic education.

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Additional info for Algebra and Number Theory: An Integrated Approach

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Conversely, if g is another greatest common divisor of a and b, then by (GCD 2) d = gu and g = dv for some integers u, v. Then d = gu = (dv)u = d{vu) and 0 = d — d(vu) — d(\ — vu). It follows that either d = 0 or 1 — vu = 0. If d = 0, then by definition a = b = 0. Therefore suppose that d φ 0. Then 1 — uv = 0 and uv = 1. 1 l(viii) implies that 1 = \u\ = \v\. Hence u = ± 1 and v — =pl. In particular, g = ±d. Thus we often speak of the greatest common divisor of two integers to mean the positive integer satisfying both (GCD 1) and (GCD 2).

4. Definition. Let A — [a¡j] and B — [b¡j] be two matrices in the set Mn(M). The product AB of these matrices is the matrix C = [c¡j], whose elements are c¡j = a¡]bij + ai2b2j H h ainbnj = 2 J \

Prove that 2" > 2n + 1 for each integer n > 3. 14. Prove that 24 divides n4 + 6n 3 + l l n 2 + 6n for each positive integer n. 15. Suppose that 2" + 1 is a prime where n is a positive integer. Prove that n = 2k for some positive integer k. 16. Let n, k be positive integers. Let n — kq + r where 0 < r < k. Prove that GCD(n, k) = GCD(£, k-r). 17. Find a positive integer k such that l + 2 + - - - + f c i s a three-digit number with all digits equal. 18. The coefficient of x in the third member of the decomposition of the binomial (1 + 2x)n is 264.

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Algebra and Number Theory: An Integrated Approach by Martyn Dixon, Leonid Kurdachenko, Igor Subbotin

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