By Marlow Anderson
A part of the PWS complex arithmetic sequence, this article contains chapters on polynomials and factoring, distinct factorization, ring homomorphisms and beliefs, and constructibility difficulties and box extensions.
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Additional resources for A First Course in Abstract Algebra: Rings, Groups, and Fields
4 A first simple example of this arithmetic is in the case where m = 2. We then have only two residue classes. In fact, 2 is precisely the set of even integers and 2 is the set of odd integers. The addition and multiplication tables for Z2 are given below. 3 Properties of Modular Arithmetic It is illuminating to compare the arithmetic on Zm with that on Z. Later in the book (in Chapter 6) we will meet a common abstraction of arithmetic on Z and on Zm that will enable us to pursue this general question in more detail.
If f has no roots, the theorem is proved. So, assume f has at least one root, call it a. 3, x − a divides f . That is, f = (x − a) · g. But deg(g) = deg(f ) − 1. And so, by the induction hypothesis, g has at most n − 1 roots. Because any root of f other than a must also be a root of g, f can have no more than n roots altogether. 6 Greatest Common Divisors We now turn our attention to finding greatest common divisors of two polynomials, paralleling our development for the integers. Notice that we didn’t say the greatest common divisor.
E. Give the prime factorizations of 92, 100, 101, 102, 502, and 1002. f. Suppose that we have two line segments. One has length 11/6 units, and the other has length 29/15. What length is the longest segment that measures both? g. 4 twice. Explain the different approaches of the two proofs to finding the appropriate linear combination. Which is easier to describe in words? Which is computationally more practical? Exercises 1. (a) Find the greatest common divisor of 34 and 21, using Euclid’s Algorithm.
A First Course in Abstract Algebra: Rings, Groups, and Fields by Marlow Anderson