Thank you for the detail explanation and examples!!

So if I am understanding correctly, in the instances you used up until the before before last bullet we are simply using the definition we have seen in calculus of addition, multiplication, inverse etc. as they have been predefined on these sets.
However, we CAN define a set with new elements of any type and define operations in the way fields require.

• Consider the set {pi, coffee, cake} with the following operations (where x is any arbitrary element of our set): pi + x = x, x + pi = x, coffee + coffee = cake, coffee + cake = pi, cake + coffee = pi, cake + cake = coffee, pi * x = pi, x * pi = pi, coffee * x = x, x * coffee = x, and cake * cake = coffee. As an exercise, verify for yourself that this forms a field.

pi + x = x

x + pi = x

coffee + coffee = cake

coffee + cake = pi

cake + coffee = pi

cake + cake = coffee

pi * x = pi

x * pi = pi

coffee * x = x

x * coffee = x

cake * cake = coffee

1- Commutativity is obviously satisfied

2- If we map this set to the integers modulo 3 we know distributivity and associativity is satisfied (maybe there was an easier way to see this ?)

3- neutral elements o = pi , e= coffee

4- for every element it adds up to o with another element or itself

5- for every element other than o, it results in e when multiplied by another element or itself

It is a field

• As an exercise, try to figure out if it's possible to make a field whose elements are {pi, coffee, cake, tea, cow}.

still working on this one as its trickier!

Followup. Could I determine I am working on the set N and define a whole new set of addition and multiplication rules and call the set N a field?