I have heard of this which doesn't seem to be revolving around numbers:
- Boolean Algebra: The set of Boolean values {0, 1} forms a field known as the Boolean field. The operations are logical AND and logical OR.
But most examples are the rational, real or complex numbers. Matrices, and functions revolving around numbers, also don't count. I am imagining possibilities like data models in software or something outside of numbers entirely. Not sure if anything like this exists.
What are some key examples of fields which help stretch your mind beyond thinking about numbers? Ideally an example which would help me gain stronger intuition for the concept of fields in a generalized way.
I mean field in the algebraic structure sense, not as in "field of knowledge."
You seek intuition about fields by asking about areas of math that don't involve numbers, when it seems you are using the word "number" to mean "a complex number". So I think you want fields that are not naturally viewed as subfields of the complex numbers, rather than something without a concept like numbers at all (such as basic topology). I don't see at all how studying something without any algebraic structure could give you intuition about general fields.
Look at finite fields and fields of rational functions such as $\mathbf C(x_1,\ldots,x_n)$. Note a field of prime size $p$ can be regarded as a quotient $\mathbf Z/p\mathbf Z$, so it comes from integers even though it's not inside the integers. Do you consider that to mean the field doesn't "involve numbers"?
The subject of field theory, as a new branch of abstract algebra, was developed by Steinitz when he learned about $p$-adic numbers $\mathbf Q_p$ where $p$ is a prime number. The $p$-adic numbers are a lot more complicated and mind-bending then the fields I already mentioned, so you could look at those too. (Some people may tell you $\mathbf Q_p$ can be embedded into $\mathbf C$ by Zorn's lemma, but such an embedding can't be written down in any natural way, so for your purposes $\mathbf Q_p$ doesn't live inside $\mathbf C$.)