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I'm working through Andy Magid's book 'Lectures on Differential Galois Theory', motivated largely because the book contains what looks to be a very elegant explanation of why some integrals, e.g $\int{e^{x^2}}dx$, can't be expressed in terms of elementary functions, found on page 82, proposition 6.12. The condition follows a different line of argument to that of Liouville's theorem, being based around differential field extensions (it's not essential to the question here, but the result, using linear algebraic groups, is that a solution of a differential equation lies in an elementary field extension if the connected component of the Galois group of the extension is abelian). My question is over Magid's definition of an elementary function, namely:

"Defn 6.11 Let $C$ be an algebraically closed field with trivial derivation and let $F=C(x)$ be the field of rational functions with derivation $D(x)=1$. A differential field extension $F(a_1,...,a_n;b_1,...,b_n)\supseteq F$ is called a field of elementary functions if for $F_j=F(b_1,...,b_{j-1})$

(1) $D(a_i)\in F$ for $1\leq i \leq n$;

(2) for $1\leq j \leq m$ either $b'_j / b_j \in F_j$ or $b_j$ is algebraic over $F_j$.

An elementary function is an element of a field of elementary functions."

As Magid says just before, "Certainly we want rational functions to be elementary functions. We also want exponential functions to be elementary functions. We also want algebraic functions to be elementary. Moreover we should be able to combine these operations. The logarithm functions, namely the integrals of rational functions, should also be elementary. This inspires [definition 6.11]".

I can see how (2) allows exponential and algebraic functions and combinations of these to be called elementary, and how (1) gives us functions such as $lnx$, $ln(x^2+1)$, or $ln$ of any polynomial. My question is how we get expressions such as $ln(lnx)$, say, or $ln(e^{ix}+e^{-ix})$ if we want the log of trig functions, both of which are expressions that I assume we want to call elementary. I imagine that this is connected to why (1) only requires $D(a_i)\in F$, i.e. in terms of $F$, rather than, as I'd have thought, based on $F_j$, e.g. perhaps $D(a_i)=c'_j / c_j$ for $c_j \in F_j$.

Can anyone explain how (1) and (2) combine to give elementary functions such as $ln(lnx)$ or $ln(e^{ix}+e^{-ix})$, given that in (1) we have $\in F$?

CJO
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    This does seem to be more restrictive than the usual definitions (see for instance Rosenlicht's 1972 Monthly paper) which allow any tower of successive algebraic, logarithmic and exponential extensions. I can't see why these should be equivalent. – Angina Seng Aug 07 '20 at 08:43
  • OK - many thanks for the reply. As you say, I can't see any obvious reason why they'd be equivalent, so am very interested if anyone can give a reason why they actually are despite first appearances (or a proof that they aren't the same). – CJO Aug 07 '20 at 10:44
  • Could it be that when Liouville's theorem shows that a function can only be integrated if it can be written in the form $\Sigma c_i u'_i / u_i + v'$, the $u_i$'s are in the base field, i.e. effectively, the condition (1) refers to the $u_i$'s and condition (2) to $v$? This is not stated by Magid but he may be assuming that any reader is familiar with it? – CJO Aug 07 '20 at 14:29

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Andy Magid has answered this question to me, saying that he does not regard the definition given in 6.11 as capturing the usual concept of an elementary extension. He agrees that ln(lnx) would not count as elementary by 6.11, although it should by the 'intuitive' idea of elementary.

CJO
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