# Extension based on similarity between rational functions and rational numbers

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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I should say that most CAS (Computer Algebra Systems) are precisely based on all these extensions! See for examples Geddes' "Algorithms for computer algebra" a random extract (books.google.fr/books?id=TXfY891RiiEC&pg=PA39) or Bronstein's tutorial (www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf) – Raymond Manzoni Jan 12 '12 at 21:18
What is it that you are calling an "elementary extension of the field of rationals"? – Arturo Magidin Jan 12 '12 at 21:19
@ArturoMagidin: I realize I was being sloppy. I learned about "elementary differential extension" of a differential field from Wiki. So to make sense, we perhaps need some derivation operator on the field of rationals to turn it into a differential field? – Tim Jan 12 '12 at 21:23
@Tim: My problem is that "elementary extension" has different meanings, and the real problem is that "elementary extension" is already a term of art in Mathematical Logic. So I honestly don't know what it is you are refering to when you talk about "elementary extension of the field of rationals". Does the field $\mathbb{Q}(\sqrt{2})$ qualify? $\mathbb{Q}(\pi)$? The ring of algebraic numbers? Or do you mean something entirely different? – Arturo Magidin Jan 12 '12 at 21:34
@ArturoMagidin: Thanks for the link! I think by "elementary extension of the field of rationals", I simply meant some field extension that has the results of applying logarithm and exponential to all rationals as members, and has the maximal algebraic extension of the rationals as a subset. – Tim Jan 12 '12 at 22:25