Algebra. Abstract and Concrete by Frederick M. Goodman

By Frederick M. Goodman

This creation to fashionable or summary algebra addresses the normal issues of teams, jewelry, and fields with symmetry as a unifying subject, whereas it introduces readers to the energetic perform of arithmetic. Its obtainable presentation is designed to educate clients to imagine issues via for themselves and alter their view of arithmetic from a approach of ideas and systems, to an area of inquiry. the quantity offers abundant workouts that supply clients the chance to take part and examine algebraic and geometric rules that are fascinating, vital, and value considering. the quantity addresses algebraic subject matters, simple thought of teams and items of teams, symmetries of polyhedra, activities of teams, earrings, box extensions, and solvability and isometry teams. For these drawn to a concrete presentation of summary algebra.

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Extra info for Algebra. Abstract and Concrete

Example text

A1 ///; and so forth. The numbers a1 ; a2 ; : : : cannot be all distinct since each is in f1; 2; : : : ; ng. ak / D a1 . 14). 13). a1 ; a2 ; : : : ; ak /: Otherwise, consider the first number b1 62 fa1 ; a2 ; : : : ; ak g that is not fixed by . b2 / D . b3 / D . bl / D b1 . b1 ; : : : ; bl /: 7 A sophisticated analysis of the mathematics of card shuffling is carried out in D. Aldous and P. Diaconis, “Shuffling cards and stopping times,” Amer. Math. Monthly, 93 (1986), no. 5, 333–348. ✐ ✐ ✐ ✐ ✐ ✐ “bookmt” — 2006/8/8 — 12:58 — page 22 — #34 ✐ 22 ✐ 1.

We call a nonzero element Œa a zero divisor if there exists a nonzero element Œb such that ŒaŒb D Œ0. Thus, in Z6 , Œ4 and Œ3 are zero divisors. On the other hand, many elements have multiplicative inverses; an element Œa is said to have a multiplicative inverse or to be invertible if there exists an element Œb such that ŒaŒb D Œ1. 7. MODULAR ARITHMETIC ✐ 41 Œ1Œ1 D Œ1, Œ3Œ5 D Œ15 D Œ1, Œ9Œ11 D Œ 5Œ 3 D Œ15 D Œ1, and Œ13Œ13 D Œ 1Œ 1 D Œ1. Thus, in Z14 , Œ1; Œ3; Œ5; Œ9; Œ11, and Œ13 have multiplicative inverses.

Write out multiplication tables for Zn for n Ä 10. 9. Using your multiplication tables from the previous exercise, catalog the invertible elements and the zero divisors in Zn for n Ä 10. Is it true (for n Ä 10) that every nonzero element in Zn is either invertible or a zero divisor? 10. Based on your data for Zn with n Ä 10, make a conjecture (guess) about which elements in Zn are invertible and which are zero divisors. Does your conjecture imply that every nonzero element is either invertible or a zero divisor?