Re: JSH: Wrappers in ring of algebraic integers

On Jul 29, 10:14 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Jul 30, 3:22 pm, JSH <jst...@xxxxxxxxx> wrote:

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

An integral domain contained in the complex numbers and containing the
integers, I'm guessing on the basis of what you say below.

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,

Non-zero rational integers?

P(x) is a polynomial with integer
coefficients where P(0) is coprime to d_1 and d_2,

In the ring of rational integers?

and where f_1(0) =
f_2(0) = 0.

What's the domain of f_1 and f_2?

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).

When you say "in every known major ring", presumably you are wanting
to put some constraints on g_1 and g_2 which are somehow related to
the ring. Explain which constraints.

Nope. The ring of algebraic integers stands alone in having this
peculiar ability to block straightforward use of the distributive
property in some cases.

For example it is possible in the ring of algebraic integers with

d_1*d_2*P(x) = (f_1(x) + d_1)*(f_2(x) + d_2)

for f_1(x) to not have d_1 as a factor and f_2(x) to not have d_2 as a
factor, where you cannot name another major ring, where I think you
cannot find another integral domain where that is possible.

After arguing about this odd ability for years I've realized that I
can prove wrapper values must be used by the ring of algebraic
integers to accomplish this feat.

The proof is simple enough as I note that no values of d_1 and d_2 can
work in the ring of algebraic integers to allow d_1*g_1(x) = f_1(x)
and d_2*g_2(x) = f_2(x) to exist when f_1(x) and f_2(x) are non-
rational and the d's are not equal by absolute value, as assume they
do, then divide off d_1*d_2 from both sides to get

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

but now you can multiply by d_1* and d_2* with integer values and get
a contradiction.

So wrappers are always forced if abs(d_1) does not equal abs(d_2).


James Harris

.

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