We started talking about fields in my foundations of mathematics class, and since the symbols we are using are + and •, I keep catching myself giving them properties of multiplication and addition. For example, instinctively thinking that $0•x = 0$ without having proved it. Since the definition of a field is so broad, it occurred to me that a field could potentially be really different from what you would get from a set of numbers, addition, and multiplication, but I can't think of any. What are some examples of fields that use operations other than addition and multiplication? It would also be especially cool if the underlying set was something other than a set of numbers, but I'd be happy with just different operations.

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    $\begingroup$ I am more than sure that there are many many great duplicates for this question all over this site. $\endgroup$ – Asaf Karagila Mar 31 '15 at 22:34
  • $\begingroup$ the elements of finite fields are not quite numbers, though they are very close to being numbers. other examples can be found in p-adic completions and function fields. however in these fields the + and * operations here are not very exotic, as they are defined via more familiar operations of number arithmetic $\endgroup$ – David Holden Mar 31 '15 at 22:36
  • $\begingroup$ @David: What are "numbers"? $\endgroup$ – Asaf Karagila Mar 31 '15 at 22:37
  • $\begingroup$ I don't know any specifics, but many theorems exclude characteristic $2$ fields. My gut says strange things happen there. $\endgroup$ – Arthur Mar 31 '15 at 22:39
  • $\begingroup$ a good question, and one i find myself asking quite often, Asaf! in fact to understand it better is one of my motivations for studying mathematics! $\endgroup$ – David Holden Mar 31 '15 at 22:40

Nimbers are objects in combinatorial game theory which describe certain kinds of games related to Nim. There is a notion of addition of nimbers making them a vector space over the finite field $\mathbb{F}_2$, which is not very surprising since it can be defined in terms of bitwise XOR. What is more surprising is that there is also a notion of multiplication of nimbers making them an algebraically closed field of characteristic $2$!

Well, almost - the collection of nimbers is too big to be a set.

  • $\begingroup$ is it correct to think of sets as a proper subcollection of the collections? $\endgroup$ – David Holden Mar 31 '15 at 22:44

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