Zero of a Monic Polynomial
Date: 02/05/2003 at 10:53:37 From: Weijia Subject: Unique factorization Hello Dr. Math, I'm trying to figure out a Number Theory problem that has to do with unique factorization. I'm supposed to show that IF the reduced fraction (a/b), which is a member of set Q, is a root of C_0 + C_1*(x) + ... + C_n*(x^n), with C_k a member of set Z for all 0<=k<=n, and C_n not equal to zero, then a|C_0 and b|C_n. In particular, show that if m is an integer, then n root of m is rational iff it is an integer. More generally, a zero of a monic polynomial is irrational or is an integer. The way the problem is worded is pretty confusing, I see some correlation in the second and third part of the question, but no correlation to the first part of the question.
Date: 02/06/2003 at 03:42:29 From: Doctor Jacques Subject: Re: Unique factorization Hello, and thank you for writing to Dr Math, Question 1 ---------- First, let me point out that we must also assume that the fraction (a/b) is in lowest terms, i.e. that gcd (a,b) = 1. Let our equation be: f(x) = c_0 + c_1*x + ... + c_n*x^n = 0 where the c_i are integers and c_n <> 0 Assume that (a/b) is a root of this equation (a and b are integers, b <> 0). We can write: f(a/b) = c_0 + c_1*(a/b) + ... + c_n*(a^n/b^n) = 0 If we multiply everything by b^n, we get: c_0*b^n + c_1*a*b^(n-1) + ... +c_(n-1)*a^(n-1)*b + c_n*a^n = 0 In this equation, we only have integers. Now, notice that everything except the first term is a multiple of a. As the sum is 0, the first term, c_0*b^n, is also a multiple of a. As gcd(a,b) = 1, a has no common factor with b^n, and this shows that a must divide c_0. Using a similar argument, you should have no trouble showing that b divides c_n. I will now consider question 3 before question 2. Question 3 ---------- Let us show that a zero of a monic polynomial is irrational or is an integer. As the polynomial is monic, we have c_n = 1. Assume that the equation has a rational root, (a/b) (in lowest terms). The previous question allows you to show that b divides c_n (=1). Isn't this what we wanted to prove ? Question 2 ---------- Let us show that if m is an integer, then n root of m is rational iff it is an integer. You correctly noticed that this is connected to the previous result. In fact, you should consider the equation: x^n - m = 0 which is monic. This allows you to use the previous result. I hope this will allow you to complete the proofs. Please feel free to write back if you want to discuss this further. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/
Date: 02/07/2003 at 10:45:50 From: Weijia Subject: Unique factorization Thank you for taking the time to help me, I really appreciated it. I was confused by the sign that looks like <>, what does <> mean? Weijia
Date: 02/08/2003 at 05:32:59 From: Doctor Jacques Subject: Re: Unique factorization It means "different from" - it's not so easy to type math in a text-only window. "<>" is the symbol used in the programming language PASCAL: you'll probably also encounter "!=", which comes from the C language. We use also "<=" to means "less than or equal to." - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/
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