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 1 Define Nim-addition and Nim-multiplication on 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 over
- Letting we have and for
Show that is a field.