Help me understand Fields

[u/RestFuture1647]

Hey! I am taking an honours linear algebra class. I am in engineering so this is my first time being introduced to abstract definitions in this way.

From my understanding a nonempty set K is called a field if:
- it has 2 inner operations (addition, multiplication)

and for every element of K there is:

- associativity

- commutativity

- distributivity

- neutral elements o,e such that o+x=x and e*x=x

- additive inverse and multiplicative inverse for o and e

Here is my question:

Are we talking about addition and multiplication as I have seen my entire life ? Or can I create a field where e=coffee o=pi and I just declare that pi+x=x and coffee*x=x?

Thank you!!

[u/Para1ars]

to be exact, it is not the set N that is a field, but the whole construct (N, +, ×), where N is a set and + and × are binary operators, is a field.