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In article <1159580701.624637.269170@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<sttscitrans@xxxxxxxxx> wrote:
>
>Arturo Magidin wrote:
>> In article <1159520041.342798.14670@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>> <sttscitrans@xxxxxxxxx> wrote:
>> >
>> >Arturo Magidin wrote:
>> >> In article <1159442146.244476.203680@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>> >> <sttscitrans@xxxxxxxxx> wrote:
>
>> >> >It seems more obvious that (2, 1+sqrt(-5), -4+2sqrt(-5))
>> >> >can't contain 1 or -1.
>> >>
>> >> That's what I don't get. The third term is just the product of the
>> >> first two, minus a multiple of the first. If it is obvious that the
>> >> latter one does not, then it should be obvious that (2, 1+sqrt(-5))
>> >> does not either.
>> >
>> >(1 + sqrt(-5))^2 = -4 +2 sqrt(-5)
>>
>> I still don't get it. WHY is it that you think that it is "obvious"
>> that an ideal (a,b,b^2) does not contain 1, but that it is NOT obvious
>> that the ideal (a,b) does not contain 1. What extra information are
>> you extracting from staring at b^2 that you could not figure out from
>> looking at just a and b?
>
>(1 + sqrt(-5))^2 = -4 +2 sqrt(-5) = 2(-2 +sqrt(-5))
So?
>(1 + sqrt(-5)) must contain a "hidden" square root of 2
Now you are using fluffy language. What exactly is gained from
that? Seems like an intuitive hunch, not a mathematical
conclusion. Sure, as it turns out, (1-sqrt(-5)) = PQ, where P and Q
are prime ideals, and P^2 = (2). But this sort of argument, while
holding in this instance, would fail in many others.
>(2, 1+sqrt(-5)) can't equal (1).
>
>If r = cubrt(10), is (3, -1 + r) equal to (1) or not ?
>
>You seem to be implying that you can tell by sight
>which alternative is the case,
No. I am wondering why it is that you seem to feel it is necessary to
explicitly list b^2 in order to do so in the case of (2, 1+sqrt(-5)).
I'm not saying you're wrong. I'm wondering why you are going through
the sort of gyrations and contorsions you are going through in order
to reach the conclusion.
--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
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Arturo Magidin
magidin-at-member-ams-org
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