Re: gcd of two polynomials over Q[i]


Jeremy Watts wrote:
"Jeremy Watts" <stevie4545@xxxxxxxxxxx> wrote in message
news:frugh.16598$HV6.11142@xxxxxxxxxxxxxxxxxxxxxxx
this is carrying on from my previous post. i have attempted to find the
gcd
of the following polynomials using the "wims" site,


http://wims.unice.fr/wims/wims.cgi?session=BO38B7C34A.3&lang=en&cmd=reply&mo

dule=tool%2Farithmetic%2Fbezout.en&x1=%286%2F5%29x%5E4%2B%28-39%2F5%2B622%2F

65i%29x%5E3%2B%281422%2F65-311%2F5i%29x%5E2%2B%28-72%2B5598%2F65i%29x%2B1296

%2F13&x2=%289%2F5%29x%5E3%2B%2857%2F5-24%2F13i%29x%5E2%2B%28-24-152%2F13i%29
x%2B320%2F13i&gcdlcm=yes&eucdiv=yes&euc1=1&euc2=2


it seems to use the standard euclidean algorithm to accomplish this, but
from what i can see the gcd is :-

((8978642999775/15211817454289-11763279964575/15211817454289I)
x+(-156843732861000/197753626905757-119715239997000/197753626905757i))


but it claims that the gcd is 3/5x-8/13i (which is indeed the correct
answer)

How does it get from the answer with these horrifically high numbers to
3/5x - 8/13i ? My guess was that it divides throughout by the gcd of
the
two complex numbers ie. the one being the coefficient of x, the other
being
the constant term, hence my first post on complex gcd's.

is this correct? how does it finally arrive at 3/5x - 8/13i ?


sorry with that link you will have to paste it into your browsers address
bar, then on the wims page check the "gcd(F1,F2) and lcm(F1,F2)" and
"Successive euclidean divisions...." boxes and press compute

Apparently your link (URL) contained a session token which is
no longer cached at the website. I have no way of knowing what
your input polynomials were, or of recomputing their CGD.

However as I've noted on the other thread, normalizing the GCD
as a monic polynomial (since Q(i) is a field) gives a canonical
representative, as Waldek Hebisch elegantly did using Axiom's
factor command (below in this thread).

regards, chip

.