Re: JSH: Stop being dense
- From: "marcus_b" <marcus_bruckner@xxxxxxxxx>
- Date: 27 Sep 2006 09:42:00 -0700
willo_thewisp@xxxxxxxxxxx wrote:
Tim Peters wrote:
As best I can make out, the only specific result
of Dedekind's you ever objected to here was this one (which he did indeed
prove using the theory of ideals):
Given algebraic integers a and b, and given that a and b are
coprime in the algebraic integers, there exist algebraic integers
a' and b' such that
a*a' + b*b' = 1
Is that the result you challenge? If so, why do you challenge it? Or if
not, what /are/ you talking about?
This hardly deserves to be called a "result"; it's an immediate
consequence
of the definition. The only thing you need to use ideal theory for is
to
define "coprime" in the first place.
Not quite. An important consequence of Dedekind's work on
ideals here was that, in the ring of algebraic integers, any two
numbers have a greatest common divisor - that is, if a and b are
algebraic integers, there exist other algebraic integers r, s, and
t such that
a = r*t and
b = s*t
and if u is another algebraic integer which divides both a and b,
then u divides t. If a and b are coprime, then you can take t to
be 1.
Marcus.
.
- References:
- JSH: Stop being dense
- From: jstevh
- Re: JSH: Stop being dense
- From: Tim Peters
- Re: JSH: Stop being dense
- From: willo_thewisp
- JSH: Stop being dense
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