Monday, July 30, 2007
JSH: Wrappers in ring of algebraic integers
I've been brainstorming yet another approach to explaining how the ring of algebraic integers is different and now can explain in a rather straightforward way how certain things have to work in that ring and why, as well as why the distributive property is key.
Consider in an integral domain let
d_1*d_2*P(x) = (f_1(x) + d_1)*(f_2(x) + d_2)
where the d's are non-zero integers, P(x) is a polynomial with integer coefficients where P(0) is coprime to d_1 and d_2, and where f_1(0) = f_2(0) = 0.
In every known major ring that is an integral domain EXCEPT the ring of algebraic integers there will always exist g_1(x) and g_2(x) such that
d_1*d_2*P(x) = (d_1*g_1(x) + d_1)*(d_2*g_2(x) + d_2)
where d_1*g_1(x) = f_1(x) and d_2*g_2(x) = f_2(x).
The ring of algebraic integer must have exception cases because if, for instance, the f's are non-rational roots of a monic quadratic with integer coefficients then it is not possible in the ring of algebraic integers for one to have a prime factor that the other does not! So if the d's have differing prime factors the g's cannot exist in that ring.
So the ring of algebraic integers applies wrappers around the the d's, which I'll call w_1 and w_2, so that you have
d_1*d_2*P(x) = ((w_1*d_1)*(w_1*g_1(x)) + (w_1)^2*d_1)*((w_2*d_2)*(w_2*g_2(x)) + (w_2)^2*d_2)
where w_1*d_1 and w_2*d_2 are roots of a monic polynomial with integer coefficients.
The wrappers are forced by the inability of non-rational roots of a monic polynomial with integer coefficients to have differing prime factors.
IN my ring of objects the wrappers are units and w_1*w_2 = 1 or -1.
To emphasize how different the ring of algebraic integers is, consider trying to start with
d_1*d_2*P(x) = (d_1*g_1(x) + d_1)*(d_2*g_2(x) + d_2)
where I remind that the d's are non-zero integers, P(x) is a polynomial with integer coefficients where P(0) is coprime to the d's.
It turns out that construct CANNOT EXIST in the ring of algebraic integers if g_1(x) and g_2(x) have any non-rational values with an integer x, and the d's do not share all the same prime factors!!!
That ring specifically blocks the distributive property itself in certain instances.
Consider in an integral domain let
d_1*d_2*P(x) = (f_1(x) + d_1)*(f_2(x) + d_2)
where the d's are non-zero integers, P(x) is a polynomial with integer coefficients where P(0) is coprime to d_1 and d_2, and where f_1(0) = f_2(0) = 0.
In every known major ring that is an integral domain EXCEPT the ring of algebraic integers there will always exist g_1(x) and g_2(x) such that
d_1*d_2*P(x) = (d_1*g_1(x) + d_1)*(d_2*g_2(x) + d_2)
where d_1*g_1(x) = f_1(x) and d_2*g_2(x) = f_2(x).
The ring of algebraic integer must have exception cases because if, for instance, the f's are non-rational roots of a monic quadratic with integer coefficients then it is not possible in the ring of algebraic integers for one to have a prime factor that the other does not! So if the d's have differing prime factors the g's cannot exist in that ring.
So the ring of algebraic integers applies wrappers around the the d's, which I'll call w_1 and w_2, so that you have
d_1*d_2*P(x) = ((w_1*d_1)*(w_1*g_1(x)) + (w_1)^2*d_1)*((w_2*d_2)*(w_2*g_2(x)) + (w_2)^2*d_2)
where w_1*d_1 and w_2*d_2 are roots of a monic polynomial with integer coefficients.
The wrappers are forced by the inability of non-rational roots of a monic polynomial with integer coefficients to have differing prime factors.
IN my ring of objects the wrappers are units and w_1*w_2 = 1 or -1.
To emphasize how different the ring of algebraic integers is, consider trying to start with
d_1*d_2*P(x) = (d_1*g_1(x) + d_1)*(d_2*g_2(x) + d_2)
where I remind that the d's are non-zero integers, P(x) is a polynomial with integer coefficients where P(0) is coprime to the d's.
It turns out that construct CANNOT EXIST in the ring of algebraic integers if g_1(x) and g_2(x) have any non-rational values with an integer x, and the d's do not share all the same prime factors!!!
That ring specifically blocks the distributive property itself in certain instances.
# posted by James Harris @ 01:22 
