By Goodman F.M.

ISBN-10: 0979914205

ISBN-13: 9780979914201

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Proof. x a/Cr, where the remainder r is a constant. a/ D r. 23. ˛/ D 0. Say the multiplicity of the root ˛ is k if x ˛ appears exactly k times in the irreducible factorization of p. 24. A polynomial p 2 KŒx of degree n has at most n roots in K, counting with multiplicities. That is, the sum of multiplicities of all roots is at most n. Proof. p/. 2. 1.

The absolute value jaj of an integer a is equal to a if a is non-negative, and equal to a otherwise. Note that a D 0 if, and only if, jaj D 0. The integers, with addition and multiplication, have the following properties, which we take to be known. 1. (a) Addition on Z is commutative and associative. (b) 0 is an identity element for addition; that is, for all a 2 Z, 0 C a D a. (c) Every element a of Z has an additive inverse a, satisfying a C . a/ D 0. We write a b for a C . b/. (d) Multiplication on Z is commutative and associative.

Show that if a divides the product bx, then a divides x. Hint: Use the existence of s; t such that sa C t b D 1. 11. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides x and b divides x, then ab divides x. 12. Show that if a prime number p divides a product a1 a2 : : : ar of nonzero integers, then p divides one of the factors. 7. 13. 14. (a) 37 Write a program in your favorite programming language to compute the greatest common divisor of two nonzero integers, using the approach of repeated division with remainders.

### Algebra. Abstract and concrete by Goodman F.M.

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