Why is an algebraic field defined with $+ - * /$ rather than other operations? In mathematics, a "field" is defined as below (not a strict one):


*

*a set of elements.

*For any $2$ elements within the set, the $+ - * /$
operations will map to another element within the same set.


My understanding is:
Mankind has the intuitive sense of quantity. So we tend to abstract concrete objects into numbers. And then we can reach interesting conclusions about the objective world through operating the numbers.
My questions is:
Why are only $4$ operations picked up when defining the field? Any other options? Why didn't we pick up the root extraction or power operation for example?
 A: I think you're looking at it backwards. You make it sound as if an entity of field has always existed, people just chose what operations to populate it with. While in reality the algebraic structures with 4 operations mentioned popped up in so many places in mathematics that it warranted for such an algebraic structure to have a name - thus the field was born. Now why would an algebraic structure with these 4 operations pop up in different branches of math and why would it need to have a name? To answer that question, I wouldn't look at the field as an entity containing 4 operations. I would look at it as one with 2 operations, that is + and *. Other two are inverse operations of + and *. Now having only 2 operations it's in a sense the second simplest algebraic structure there is, the only simpler one would be group, that is a structure with a single operation.
A: The Greeks implicitly considered the field of constructible numbers, which is the smallest set of numbers closed under the ordinary arithmetic operations and square roots; it is the quadratic closure of the rational numbers.
So, this earliest example of field does indeed consider other operations.
