Lindsay N. Childs's A Concrete Introduction to Higher Algebra PDF

By Lindsay N. Childs

ISBN-10: 1468400657

ISBN-13: 9781468400656

ISBN-10: 1468400673

ISBN-13: 9781468400670

This booklet is a casual and readable creation to better algebra on the post-calculus point. The recommendations of ring and box are brought via learn of the commonplace examples of the integers and polynomials. the hot examples and thought are in-built a well-motivated style and made appropriate through many functions - to cryptography, coding, integration, heritage of arithmetic, and particularly to undemanding and computational quantity thought. The later chapters contain expositions of Rabiin's probabilistic primality try out, quadratic reciprocity, and the class of finite fields. Over 900 workouts are chanced on through the book.

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Additional info for A Concrete Introduction to Higher Algebra

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Basic Properties The congruence symbol looks like an equality, and this is no accident. In fact, one can view congruence geometrically as a kind of equality. Take the real number line and lay on it a circle of circumference n. Then wind the real number line around the circle. The picture shows n = 6. 3,9,15 -3 17,11,5 -6 1,7,13,19 0,6,12,18 Integers which lie over the same point on the circle are congruent mod n; integers lying over different points are not. Most of the things you normally can do with an equality you can do with congruence mod n.

P:', b = p{1 ... pf. Then in the equation 2a 2 = b 2 , 2 occurs to an even power on the right side, and to an odd power on the left side, which is impossible. 0 PROOF. E28. Prove: If (a, b) = I and ab is a square, then a is a square. E29. Prove: If d=(a, b), then {ax + bylx,y in Z} = {zdlz in Z}. E30. If (a, b) = 8, what are the possible values of (a 3, b4)? E31. If (a, b) = p3,p prime, what is (a 2, b2)? E32. Prove by induction that if a prime number p divides a)· a2 ... an> then p must divide one of the factors aj.

Since r' - r;;;' 0 and a > 0, q - q' ;;;. o. Also, r' - r ~ r' < a, so a( q - q') < a, so q - q' < 1. Since q - q' is an integer, we must have q - q' = 0, and q = q'. Hence also r' - r = a(q - q') = 0, and r' = r. That completes the proof of uniqueness, and of the division theorem. 0 We proved the existence part of the division theorem by well-ordering in Chapter 2. Here is some terminology about integers: Say that a divides b, if b = aq for some irlteger q. We shall write alb if a divides b. For example, 6 divides 12, 6 doesn't divide 15.

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A Concrete Introduction to Higher Algebra by Lindsay N. Childs

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