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 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