This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 55  Homework 6 Solutions Friday, July 22, 2011 1. Polynomials mod p . In this problem: p is a prime. all polynomials have integer coefficients. Definitions: For polynomials f ( x ) = a k x k + ... + a 1 x + a and g ( x ) = b k x k + ... + b 1 x + b , we say that f ( x ) g ( x )(mod p ) if for every i , a i b i (mod p ). For a polynomial f ( x ) = a k x k + a k 1 x k 1 + ... + a x + a , define deg p ( f ) to be the largest integer (at most k ) so that p 6  a deg p ( f ) (and deg p ( f ) = if no such integer exists.) (a) If f ( x ) g ( x )(mod p ), and m is an integer, prove that f ( m ) g ( m )(mod p ). Let f ( x ) = a k x k + ... + a 1 x + a and g ( x ) = b k x k + ... + b 1 x + b . By assumption for every i , a i b i ( mod p ) . Now f ( m ) = a k m k + ... + a 1 m + a b k m k + ... + b 1 m + b = f ( m )( mod p ) as desired. (b) Prove the for any polynomial f ( x ), there is a polynomial g ( x ) of degree deg p ( f ( x )) so that g ( x ) f ( x )(mod p ). Let f ( x ) = a k x k + ... + a 1 x + a and let deg p ( f ( x )) = j k . So a i 0( mod p ) for i > j . And a j 6 0( mod p ) so a j 6 = 0 . Put g ( x ) = a j x j + ... + a 1 x + a . (If j = then g ( x ) = 0 .) So the degree of g ( x ) is deg p ( f ( x )) by how it is defined. For i > j , a i 0( mod p ) . and for i j , a i a i ( mod p ) . So f ( x ) g ( x )( mod p ) . (c) Prove that if f ( x ) and g ( x ) are polynomials, then deg p ( f ( x ) g ( x )) = deg p ( f ( x )) + deg p ( g ( x )). Using part (b), replace f and g with e f and e g whose degrees are deg p ( f ) and deg p ( g ) respectively. Then e f ( x ) e g ( x ) has degree deg p ( f ) + deg p ( g ) . Also, the leading coefficients of e f ( x ) and e g ( x ) are not divisible by p , so neither is the lead ing coefficent of e f ( x ) e g...
View
Full
Document
This note was uploaded on 03/14/2012 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.
 Summer '08
 STRAIN
 Polynomials

Click to edit the document details