Recall that we defined Nim addition and Nim multiplication on the natural numbers by setting:
- is the result of writing in binary and adding without carrying.
- is the unique binary operation on that is commutative, associative, distributive over and satisfies:
- whenever . Here, for all , call these numbers “Fermat’s 2-powers”.
The first problem is that it is not quite clear that is even well-defined (i.e., are there any functions at all that behave as required of ? Is there really only one such function?). Begin by checking this, by showing that the rules above give us that is the result of writing as sums of (multiplications by powers of 2) of Fermat 2-powers, and expanding using associativity, distributivity, and the two rules about how to multiply with Fermat 2-powers. (Verify that any can indeed be written as such a sum in a unique way, and that this indeed shows that is completely characterized by our description.)
Show that if we set , then and each are fields (over , with addition and multiplication given by .
In lecture I erroneously mentioned that is algebraically closed. This is not the case. For example, show that the equation has no solutions in when we interpret “ is a solution” to mean that .
Nim addition and multiplication were introduced by John Conway. A bit of online searching will give you references for this exercise, but please abstain from looking for them. I will provide references and some additional details once the homework has been turned in.
This set is due February 11 at the beginning of lecture.