Re: Algebraic integers

From: *** T. Winter (***.Winter_at_cwi.nl)
Date: 12/12/04 

Date: Sun, 12 Dec 2004 02:37:28 GMT

In article <cpg105$2mu2$1@agate.berkeley.edu> magidin@math.berkeley.edu (Arturo Magidin) writes:
> 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.

The usual norm, for quadratic fields.

> 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.

Yes, in some cases such a norm (as Hardy & Wright call it) will not
result in algebraic integers. However, at the level the students
apparently are, they would have difficulty proving that it is *not*
an algebraic integer. They have just touched the Gaussian integers.
Nevertheless, while they are not algebraic integers, they are integer-like,
and that was the purpose; to show that there were interesting other
integer-like rings.

But I was indeed wrong when I said that they could prove that the
"integers" so defined were algebraic integers. I shouls really pick
up again Hardy & Wright to see how they do it. 

-- 
*** t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~***/