I refer to the textbook for the basic notion and properties of *vector spaces*. A *field* is a triple satisfying the axioms listed in pages 2, 3 of the textbook as properties of see also https://andrescaicedo.wordpress.com/2009/02/11/305-4-fields/ and surrounding lectures, in this blog. I am writing this note mostly to record the exercise, the question, and the statement of Steinitz lemma, so I am not recording the proofs we discussed in lecture.

The following example, that I want to leave as a (voluntary) exercise, is due to John Conway.

Exercise 1DefineNim-additionandNim-multiplicationon as follows:

is the result of adding without carrying the binary expansions of and . For example,is computed by applying the following rules:

is commutative.distributes overLetting we have and for

Show that is a field.