Re: Algebraic integers

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 12/11/04 

Date: Sat, 11 Dec 2004 23:48:21 +0000 (UTC)

In article <I8JAtw.MzB@cwi.nl>, *** T. Winter <***.Winter@cwi.nl> wrote:

>I honestly think that the best way to put exercises to the students is
>to consider other quadratic fields, with a norm function and defining
>integers to be those that have integer norm.

What norm? Certainly not the usual norm.

For example, in Q(sqrt(-15)), the norm of (1/4) + (1/4)*sqrt(-15) is
1, an integer, yet it is clearly not an algebraic integer.

In fact, any root of an irreducible polynomial of the form x^2 + ax +
b, with b an integer, will have integer norm but will only be an
algebraic integer if a is also an integer.

>This results in interesting
>division properties. It also clearly shows that the rings of those
>"integers" are integer-like, but sometimes also have strange properties.
>And they can at least prove that "integers" so defined are indeed
>algebraic integers (but not the other way around, and this is what you
>seem to wish).

Ah; so you're dealing with an order rather than with the ring of
algebraic integer. Okay then.... 

--
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@math.berkeley.edu
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