Is rational number system the motivation behind the definition of field structure?

Question

It might sound a very silly question but I didn't find satisfactory answer anywhere. When I read concept of order(I am reading baby Rudin now), I find it very intuitive in sense that it connects to real world arrangement of object. Meaning if we had collection of objects, what we naturally do is arrange them one after another. That's exactly what the definition of order does with set. Also I find definition of metric space very much connected to real world in sense of distance. So on basis of what I have read so far, I find that, we take a set and attach some notion to it and thus convert it to some structure. I am curious to know how that particular notion came, what's the idea behind it, like I said about notion of order and distance function in metric space. Now when I look back at the definition of field, I thought for a long time about how someone came up with notion of addition, multiplication, element 0, element 1, additive and multiplicative inverse.. there's so much. What was motivation behind defining this all notions. Then I felt that it might be the case that we were already familiar with the rational number system before the birth of 'field structure'. And taking intuition from the properties that rationals posses, we made precise definition of field. But this all are my thoughts. Unknowingly I am getting habit of finding intuition or motivation behind every definition, which I feel is not good always. Thanks.

Answer 1

You are asking about the intuition and motivation for the definition of a field. It seems to be the case that finite rings and fields led to the definition. Early in the 1800s Gauss developed the idea of the congruence relation of integers. This implicitly defined the ring of integers modulo $\,n\,$ which is a finite ring. When $\,n\,$ is a prime it is also a finite field. A little later Galois extended this by introducing finite fields with prime power number of elements. This led to the idea of algebraic extensions of existing fields such as the real number field. These important special cases motivated the definition and study of fields.

The Wikipedia article Fields has more details about the history and development of field theory going back before 1800.