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.


James Harris

.

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