From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Help with some Algebra Problems?
Date: 6 Nov 1999 07:24:59 GMT
Newsgroups: sci.math
Keywords: sums of squares in  Z[i]

A week's gone by so perhaps it's OK to look at these:

In article <381B57A7.11947A31@zeno11.math.uwaterloo.ca>,
Andy Jay Kwak  <andy.kwak@canada.com> wrote:

>For the Ring of Gaussian Integers,  Z[i] prove
>1.  A Gaussian integer which is the sum of squares in Z[i] is the sum of
>    3 squares in Z[i]
>2.  prove the 2+2i is the sum of 3 squares but not the sum of  squares
>    in Z[i]

Every square in  Z[i]  clearly has an even imaginary part, so sums must also. 
On the other hand, observe that  a^2+b^2 = (a+ib)(a-ib), so that conversely
xy = ((x+y)/2)^2 + (i(y-x)/2)^2 = (x + (y-x)/2)^2 + (i(y-x)/2)^2 
that is, every product is a sum of two squares if the two factors are congruent
mod 2. So every Gaussian integer of the form  (2n+1) + (2m) i  is 
surely a sum of two squares, and so every integer of the form  2n + 2m i 
is a sum of three squares. If  2 + 2i = (a+ib)(a-ib), then  a+ib  would 
have to be one of the factors of  2+2i = -i (1+i)^3 ...

>Prove that if F is a field that F[[x]] (ring of formal power series) is
>a Euclidean Domain

Even better (and probably just as easy to prove) every element of  F[[x]]
has a unique representation   u x^n  for some non-negative integer  n
and some unit  u  of  F[[x]].

>(AK - how is multiplication defined for F[[x]]?  )
By collecting like powers of  x .

dave