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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.

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    $\begingroup$ You might be looking for structures isomorphic to a certain structure. That way, they may be considered the same. $\endgroup$ Commented Jul 9 at 5:41
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    $\begingroup$ The precise question is pretty unclear to me, but I'll say one interesting source of fields comes from constructible numbers. This line of thought is how the impossibility of general angle trisection is proved. The fact that the constructible numbers form a field with specific algebraic properties ends up being absolutely crucial. $\endgroup$ Commented Jul 9 at 6:04
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    $\begingroup$ Yes, recognizing that the constructible numbers form a field is the first key insight in resolving a problem that was open for thousands of years. It's considered pretty basic nowadays with the benefit of hindsight and the development of vastly more complicated algebraic machinery, but it's still an achievement. $\endgroup$ Commented Jul 9 at 6:09
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    $\begingroup$ google "structuralism mathematics axiomatic groups rings fields", then read the "AI overview". $\ \ $ $\endgroup$ Commented Jul 9 at 6:21
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    $\begingroup$ Quite the opposite to finite cases, knowing a infinite set with two specific operations is a field doesn’t say much as far as I am concerned. There are so many different fields! The meromorphic functions on a Riemann surface form a field. It’s hard to say we can read off much information from this observation. However, it does suggest one direction is to study it via some generic field-theoretic methods, for instance to look at its absolute Galois cohomology. Such endeavors are often difficult and esoteric, I’d wager. $\endgroup$ Commented Jul 31 at 6:47

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