I know that the algebraic structure represented by groups, rings, and fields is a great help in mathematical research because it abstracts the operation structure of number systems such as real numbers $\mathbb{R}$, rational numbers $\mathbb{Q}$, etc., so when an unknown mathematical object appears, if it matches this algebraic structure, it can be known that it is an equivalent system to that system.
However, for example, the structure called field also includes various number systems such as real numbers $\mathbb{R}$, rational numbers $\mathbb{Q}$, and complex numbers $\mathbb{C}$, and the sets of real numbers, rational numbers, and complex numbers that I wrote above all mean that the structure is similar as a field, but they are not exactly the same. So I am posting this question because I am curious whether the motivation of the algebraic structure that I wrote above is a concept that is also applicable in mathematical research.
So I actually want to know an actual case in which, in mathematical research, the algebraic structure is the same and it is concluded that it is the same as the set of the corresponding number system.
In conclusion, for example, even if an unknown object matches an algebraic structure called a field, we cannot know in detail whether the object is the same as a real number, a rational number, or a complex number, but I want to know how, in mathematical research, we can find out what the mathematical meaning of the unknown object is just by the fact that it is a set equivalent to the field.