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I am unable to find a reference where the definition of algebraic functions is given. I am looking for one.

For example, I see in many lectures that the function $\left\{\begin{array}{crcl}f:&\mathopen[0,+\infty\mathclose[&\to&\mathbb R\\&x&\mapsto&\sqrt{x}\end{array}\right.$ is an algebraic function. But for me, algebraic means "over a field" and since the algebra of functions $\left\{g:\mathopen[0,+\infty\mathclose[ \to\mathbb R\right\}$ is not integral, it can not be endowed in a field. So talking about algebraic function is misleading for me.

That's why I am looking for a precise definition of algebraic function.

J. W. Tanner
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joaopa
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    The terminology refers to functions that are algebraic over the embedded constant field of rational functions. For example, $f$ is algebraic because it satisfies $f(x)^2 - x = 0$. – LSpice Sep 10 '23 at 20:03
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    (Typo: "the field of rational functions"; "embedded constant" does not belong.) Re, once you consider functions with domains on, say, different intervals, you start losing nice algebraic properties (what is the domain of $\sqrt{x - 1} + \sqrt{-2 - x}$?), so it may not be so useful to think of the resulting structure as a field. Your TeX does not seem to render as intended, but, no, we do not require that rational functions be everywhere defined. – LSpice Sep 10 '23 at 20:10
  • So i do not still understand. Is the function $\left{\begin{array}{crcl}f:&\mathbb R&\to\´\mathbb R\&x\mapsto&\begin{cases}e^x&\text{if }x<0\\sqrt x&\text{if }x\ge 0}\end{array}\right.$ algebraic?. With your definition one can find a smaill set where it is, but globally it is not. – joaopa Sep 10 '23 at 20:21
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    Re, I agree. I am only used to seeing the term in calculus textbooks, where algebraic subtleties tend to be less regarded. If I were to see the term in a more advanced mathematical context, then I would expect to see more details—and, of course, any proofs would give a hint of what was meant. Where are you seeing the term? – LSpice Sep 10 '23 at 20:24
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    FWIW, one can easily give a precise definition for what it means for a formal power series to be algebraic, so that $(1+x)^{1/2}$ is algebraic (though not $x^{1/2}$, which doesn’t make sense as a formal power series). – Sam Hopkins Sep 10 '23 at 22:11
  • Read here. Is this discussion sufficiently rigorous? The trick is to think not of a function of one variable but of a rational function of several variables restricted to a suitable affine algebraic curve. Ditto for algebraic functions of several variables. – Moishe Kohan Sep 19 '23 at 04:08
  • Relevant: The MSE question Precise definition of an "algebraic function". Among other things of possible interest there, see the paper I cite in a comment to GEdgar's answer. (Site operation note: I remembered writing something to this effect in a comment, looked through my recent comments in mathoverflow and found it, but when I click on the question itself my comment no longer appears. However, I was able to copy the comment from my list of recent comments. If my comment was deleted for some reason, (continued) – Dave L. Renfro Sep 19 '23 at 07:34
  • then please note that the MSE question I cited is nearly (if not actually) a duplicate and both GEdgar's answer there and my comment there are particularly relevant to the present question. The reason I gave this in a comment is because link-only answers are strongly discouraged and I don't have the time or interest now (or 6 days ago when I wrote my apparently deleted comment) to properly write an answer. Incidentally, my happening to remember that over 5 years old MSE question and locating it was a nontrivial task, and having that work erased by a comment deletion is a bit irritating.) – Dave L. Renfro Sep 19 '23 at 07:39
  • @DaveL.Renfro, re, is the comment you mean this one? It still shows up for me. I copy the reference from there: "Fred Richman, Algebraic functions, calculus style, Communications in Algebra 40 #7 (2012), pp. 2671-2683 (27 July 2010 preprint version)." – LSpice Sep 30 '23 at 02:19
  • @LSpice: No, I did not mean the comment to GEdgar's answer from over 5 years ago, but instead comments to THIS question that were made before this question migrated from mathoverflow to MSE (note I said "... 6 days ago when I wrote my apparently deleted comment"). Regarding your re link, the URL sent me to a page saying (in part) "Page not found We're sorry, we couldn't find the page you requested." Incidentally, if this message arises due to my not having an appropriate reputation (continued) – Dave L. Renfro Sep 30 '23 at 04:50
  • or some appropriate administration privilege, then the message should probably say this. There is a big difference between not being able to find a certain page (e.g. a dead link, URL incorrectly typed at very end somewhere, etc.) and failing to have the appropriate permission. Of course I realize that probably no one reading this, even a MSE moderator, has any authority for something like this. – Dave L. Renfro Sep 30 '23 at 04:50

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