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/