Monday, July 30, 2007

 

JSH: Weird but fascinating

Whew! Quite a bit of work from yesterday to today when I finally realized I was brainstorming out a proof that I now call the wrapper theorem. Turns out that the key to all that freaking arguing over the years was noting that if you have a polynomial with integer coefficients in an integral domain then the factorization

P(x) = (g_1(x) + 1)*(g_2(x) + 1)

is BLOCKED when with non-zero integer x, the g's are non-rational, which is just this remarkable, odd little result that NO ONE would notice unless they went looking for it, as you cannot see it with polynomial factorizations, of course.

So now you get this odd thing with the distributive property as then

p_1*p_2*P(x) = (p_1*g_1(x) + p_1)*(p_2*g_2(x) + p_2)

where the p's are differing prime numbers is blocked from the ring of algebraic integers as well, when the g's are non-rational with non-zero rational x.

But it gets weirder!!!

Because you CAN have the factorization

p_1*p_2*P(x) = (f_1(x) + p_1)*(f_2(x) + p_2)

in the ring of algebraic integers with the f's being non-rational with non-zero rational x.

And you can because you can keep substituting to get to

p_1*p_2*P(x) = (h_1(x) + p_1*p_2)*(h_2(x) + p_1*p_2)

and get the symmetry that the ring of algebraic integers is requiring if the functions are non-rational with rational non-zero x.

As then the h's can be algebraic integer functions, being roots of a monic polynomial expression with integer coefficients as I've often demonstrated with what I call non-polynomial factorization!

What a story! What a wacky ring! It is so cool. So oddly quirky.

No wonder mathematicians like the ring of algebraic integers so much, it's almost human in its peculiarities!!!





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