### Tuesday, January 06, 2009

## JSH: General distributive property result, helps?

There is a general result on the complex plane that with n, a natural number, given

n*P(x) = (n*f_1(x) + n)(f_2(x) + 2)

where P(x) is a polynomial and

f_1(0) = f_2(0) = 0,

you have a demonstration of the distributive property.

Notice it also follows then that with

n*P(x) = (f_1(x) + n)(f_2(x) + 2)

with

f_1(0) = f_2(0) = 0

you do as well.

It may seem trivial but it's a fairly powerful result if you pick a ring where the f_1(x) and f_2(x) exist, as there you have a factor result!!!

That is, given

n*P(x) = (f_1(x) + n)(f_2(x) + 2)

and f_1(0) = f_2(0) = 0

in, say, the ring of integers, you have that f_1(x) must have n as a factor.

That is true for the ring of integers, and for gaussian integers, but it is not true for the ring of algebraic integers, which is why math people fight that result, which is the generalization of the construction I've been using with n=7.

Here I'm not going to make it as easy for posters disagreeing with me.

If they dispute the result, then give a counterexample in some ring other than the ring of algebraic integers.

Like pick n=32 and show f_1(x) coprime to 2 or something, or do something with gaussian integers.

But when you fail to refute this result, act like you care at all about mathematics and just admit you are wrong.

n*P(x) = (n*f_1(x) + n)(f_2(x) + 2)

where P(x) is a polynomial and

f_1(0) = f_2(0) = 0,

you have a demonstration of the distributive property.

Notice it also follows then that with

n*P(x) = (f_1(x) + n)(f_2(x) + 2)

with

f_1(0) = f_2(0) = 0

you do as well.

It may seem trivial but it's a fairly powerful result if you pick a ring where the f_1(x) and f_2(x) exist, as there you have a factor result!!!

That is, given

n*P(x) = (f_1(x) + n)(f_2(x) + 2)

and f_1(0) = f_2(0) = 0

in, say, the ring of integers, you have that f_1(x) must have n as a factor.

That is true for the ring of integers, and for gaussian integers, but it is not true for the ring of algebraic integers, which is why math people fight that result, which is the generalization of the construction I've been using with n=7.

Here I'm not going to make it as easy for posters disagreeing with me.

If they dispute the result, then give a counterexample in some ring other than the ring of algebraic integers.

Like pick n=32 and show f_1(x) coprime to 2 or something, or do something with gaussian integers.

But when you fail to refute this result, act like you care at all about mathematics and just admit you are wrong.