The reference is Chapter 6 of ONAG (in particular Thm 49). The
terminology
of "nimbers" appears in "Winning ways for your mathematical plays".
Michel
On Mar 20, 8:11 am, "Michel" <[EMAIL PROTECTED]> wrote:
> 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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