Monday, August 23, 2010


JSH: Weirdness with ring of algebraic integers

Oddly enough I've found a simpler demonstration of the weirdness of algebraic integers, as consider the following is NOT valid in that ring:

7(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 7)(f_2(x) + 7) = 7*P(x)

if P(x) is a polynomial irreducible over Q. Here note the f's and g's are blocked from being algebraic integer functions!!!

When I trotted this out a couple of days ago I also thought you needed g_1(0)=g_2(0)=0, but now don't think that is the case.

So bizarrely enough that seemingly simple algebraic expression is not available in the ring of algebraic integers.

Here arguments are unnecessary, as all anyone has to do is give valid algebraic integer functions for the f's and g's.

It's a stunning demonstration: the ring of algebraic integers actually stops algebra itself in this peculiar way.

It tells Mother Algebra—thou shalt not do that here.

<< Home

This page is powered by Blogger. Isn't yours?