By Lindsay N. Childs

This booklet is a casual and readable advent to better algebra on the post-calculus point. The techniques of ring and box are brought via research of the customary examples of the integers and polynomials. the hot examples and conception are in-built a well-motivated model and made proper through many purposes - to cryptography, coding, integration, historical past of arithmetic, and particularly to common and computational quantity thought. The later chapters contain expositions of Rabiin's probabilistic primality try, quadratic reciprocity, and the class of finite fields. Over 900 workouts are came across in the course of the book.

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**Extra resources for A Concrete Introduction to Higher Algebra**

**Example text**

3 . 3 . 31 and 2 . 3 . 2 . 31 are different. Factorization of a is unique if any two factorizations of a are the same. Theorem. Any natural number ~ 2 factors uniquely into a product of primes. PROOF BY INDUCTION (2). Suppose that the result is true for all numbers less than a. Suppose a = PI ... Pn and a = ql ... qm are two factorizations of a. We want to show that the two factorizations are the same. Suppose it is true that PI = f/j for some j. Then a/PI = P2 ... Pn = ql ... f/j-lqj+1 ... qm' Since a/PI

B. Some Analytic Results This section assumes some knowledge of calculus and of infinite series. It and the following section may be omitted without loss of continuity. Euler provided a proof of the existence of infinitely many primes which provides the starting point for much advanced number theory. n = E lim 00. E-+oo n= I PROOF OF EULER. As in Euclid's proof, we assume that there are only finitely many primes, and we derive a contradiction. If PI' ••• ,PN are all the primes, then the product g(1- /1/ N Pi) ) is finite.

Theorem. If rN is the last nonzero remainder in Euclid's algorithm for a and b, then rN is the greatest common divisor of a and band rN = ax + by for some x,y. PROOF. If rN is the last nonzero remainder in Euclid's algorithm for a and b, then the number of steps in the algorithm is N + 1. We shall prove the theorem by induction on N. If N = 0, then a divides b, and the theorem is trivial. If N = 1, then Euclid's algorithm for a and b has the form: + r l, r l q2 + o. b = aql a = Then it is easy to see that r l is the greatest common divisor of a and b; also r l = b· 1 + a· (- ql), so Bezout's identity holds.