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After realizing the similarity between rational functions as to polynomial functions and rational numbers as to integers, and the similarity of algebraic function to algebraic numbers, I am tempted to draw a parallel comparison between further generalizations from them, although I am not sure if it will be meaningful and helpful for me to learn some new concepts from you guys:

  • the ring of polynomials VS the ring of integers;
  • the field of rational functions VS the field of rationals;
  • the elementary extension of the field of rational functions VS the elementary extension of the field of rationals
  • the extension of the field of rational functions wrt some order or metric (?) VS the field of reals
  • the algebraic extension of the above VS the field of complex numbers

My questions are:

  1. Is the elementary extension of the field of rationals a proper subset of the set of complex numbers? Do people study it?
  2. Do people study the extension of the field of rational functions wrt some order or metric, just as the field of reals being the extension of the field of rationals wrt its order or metric?
  3. If yes, do people study the algebraic extension of the above, just like the field of complex numbers being the algebraic extension of the field of reals?

Thanks and regards!

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    @RaymondManzoni: Thanks for the links!2012-01-12

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