Yes the basic reference is ONAG. I will put a more precise
reference in the source (but I need to go to the library
to fetch ONAG). Basically the two rules for multiplying
nimbers are
(1) The product of any number of distinct Fermat powers
is the ordinary product.
(2) If f is a Fermat power then f^2=3f/2 (where 3f/2 is
computed for the ordinary product).
Basically (1)(2) say that if f_n is the n't Fermat power then
f_n^2=f_n+f_{n-1}f_{n-2}...f_0
This realizes the nimber field as a tower of quadratic extensions.
To compute in the nimber field we write nimbers < f^2 (f a Fermat
power) as l*f+r with l,r<f and work recursively.
Michel
On 19 mrt, 21:29, "David Joyner" <[EMAIL PROTECTED]> wrote:
> Are you using ONAG for the main reference? In any case, I would appreciate
> a precise reference to a book or article on nimbers.
>
> On 3/19/07, Michel <[EMAIL PROTECTED]> wrote:
>
>
>
> > Hi,
>
> > To acquant myself with sage's inner workings I have implemented
> > Conway's nimber field.
> > See
>
> >http://alpha.uhasselt.be/Research/Algebra/Members/nimbers/
>
> > Recall that the nimbers form a field whose underlying set is the
> > natural numbers. The addition is bitwise exclusive or but the
> > multiplication is complicated. GF(2^(2^n)) is isomorphic to the
> > nimbers that are less than 2^(2^n). Thus the full nimber field is
> > isomorphic to the union of GF(2^(2^n)) for all n.
>
> > Although my implenentation is still in pure python it seems to be not
> > much slower
> > than the standard finite fields GF(2^(2^n)) that one can create in
> > sage. However I didn't
> > do extensive testing. The basic arithmetic should be trivial to
> > rewrite in pyrex.
>
> > This is still a prototype. The most glaring ommission is that
> > coercions from and to
> > standard Galois fields are missing. Nevertheless if there are remarks/
> > comments I would
> > appreciate it very much.
>
> > Regards,
> > Michel
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