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# Greatest common divisor of two polynomials question

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1. Greatest common divisor of two polynomials question
A part on wikipedia doesn't make sense to me;

http://en.wikipedia.org/wiki/Greates...wo_polynomials

Let and be non-zero polynomials, with coefficients in a field . A greatest common divisor of and is a polynomial that is a divisor of and of .

Note: If is another polynomial, then it too is a greatest common divisor of and if and only if is equal to multiplied by an element of .

I don't understand how this last statement (the 'note') can be true. For example take (the field F is the real numbers);

and

In this case . From the last statement does it not say that is also a greatest common divisor? It's clearly not though.

That source was from wikipedia, my lecture notes say it in a different way;

Also if is another polynomial that divides and then divides . Write . This polynomial is defined only up to a non-zero scalar multiple so, if we want a unique gcd, then we insist be monic (the co-efficient of the highest power of is 1).

I don't get how make the co-efficient of the highest power makes the gcd unique. Surely, by the conditions imposed on the gcd, the gcd would always be unique anyway regardless of th co-efficient in front of the highest power?
Last edited by Sasukekun; 19-04-2012 at 14:15.
2. Re: Greatest common divisor of two polynomials question
(Original post by Sasukekun)
...
Using the example you've given:

d(x) = ax, where a is any non-zero real.

I.e. are all gcd's of your two functions.

But there is only one monic gcd, and that is "x".
3. Re: Greatest common divisor of two polynomials question
It's because 2 is a unit in the field of real numbers. The gcd is not unique when you're talking about the integers: is the gcd of 6 and 15 equal to 3 or -3? It doesn't really matter. if d and d' are two greatest common divisors (over an integral domain) then we obtain d | d' and d' | d. So if d = d'a and d' = db then ab = 1, so a and b are units, meaning d and d' are associates.
Last edited by Glutamic Acid; 19-04-2012 at 14:49.
4. Re: Greatest common divisor of two polynomials question
(Original post by ghostwalker)
Using the example you've given:

d(x) = ax, where a is any non-zero real.

I.e. are all gcd's of your two functions.

But there is only one monic gcd, and that is "x".
(Original post by Glutamic Acid)
It's because 2 is a unit in the field of real numbers. The gcd is not unique when you're talking about the integers: is the gcd of 6 and 15 equal to 3 or -3? It doesn't really matter. if d and d' are two greatest common divisors (over an integral domain) then we obtain d | d' and d' | d. So if d = d'a and d' = db then ab = 1, so a and b are units, meaning d and d' are associates.

Thanks for the replies. I've understood what you guys have said but apologies as I am slow at understanding this. So in my example the field was the real numbers. When we define gcd, in this case, it does not matter that the GCD does not divide the numbers evenly? Ghostwalker, you for example said is a GCD of those two functions.

But

and

Is a GCD because the polynomial left behind still has a co-efficient in the real numbers (i.e. ?
Last edited by Sasukekun; 19-04-2012 at 18:25.
5. Re: Greatest common divisor of two polynomials question
(Original post by Sasukekun)
Thanks for the replies. I've understood what you guys have said but apologies as I am slow at understanding this. So in my example the field was the real numbers. When we define gcd, in this case, it does not matter that the GCD does not divide the numbers evenly? Ghostwalker, you for example said is a GCD of those two functions.

But

and

If the field was the integers, this would not work because when taking a factor of out it does leave behind a polynomial that has an integer co-efficient. Right?
The set of integers forms a ring, not a field; and in a general ring, gcds needn't exist.
6. Re: Greatest common divisor of two polynomials question
Of course my mistake, I'll edit out that last bit then just to avoid confusion with my question.
7. Re: Greatest common divisor of two polynomials question
(Original post by Sasukekun)
Is a GCD because the polynomial left behind still has a co-efficient in the real numbers (i.e. ?
Yes. 1/pi is just a scalar, and it's valid as you're working over R.
8. Re: Greatest common divisor of two polynomials question
Because any two GCDs divide each other, the GCD is unique up to multiplication by a constant - so to make thing simpler we can just define "the GCD" to be the monic GCD (which is unique). Just like with integers it's unique up to sign - so we can just define "the GCD" to be the positive one.

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Last updated: April 19, 2012
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