XPost: sci.math   
   From: ivgroups@onlinehome.de   
      
   "Waldek Hebisch" wrote in news:mspfu2$4sc$1@z-news.wcss.wroc.pl...   
   >>> 1)   
   >>> We know that some polynomials have solutions, but their   
   >>> Galois groups are not solvable. Therefore there are algebraic   
   >>> numbers which cannot be explicitly described by an algebraic   
   >>> expression.   
   >>> 1a) Is there a name for this kind of algebraic numbers?   
   >>> 1b) Is there a method to construct or describe this kind of   
   >>> algebraic numbers only by symbolic methods, e.g. by closed-   
   >>> form expressions?   
   >>> 2)   
   >>>          [the same as 1), but "algebraic functions" instead of   
   >>>           "algebraic numbers"]   
   > For subpoint b: there   
   > is easy to represent roots of irreducible polynomial P in symbolic   
   > way.  Just introduce a new name (say b) for a root and ...   
   Is this polynomial factorization?   
      
   >>> 3)   
   >>> Liouville and Ritt ("Integration in finite terms") define the   
   >>> class of elementary functions by algebraic equations over   
   >>> algebraic or transcendent monomials. Considering 1) and   
   >>> 2), this means that their class of elementary functions   
   >>> contains also the functions which are not expressible by an   
   >>> algebraic expression. This contradicts the normal use of the   
   >>> term "elementary function".   
   > 3a) Am I right?   
   > Partially, but mostly wrong.   
   Which statement is wrong?   
      
   >>> 3b) Has this incorrect definition of the class of elementary   
   >>> functions somewhere any consequences?   
   > "elementary function" were defined by Liouville.  Since this is   
   > definition you can not call it wrong.  In fact, the intent was   
   > to wark with functions given in "finite terms".  Roots of   
   > polynomials are described by finite amount of data and one   
   > can do computations with them, so they are given in   
   > "finite terms".  At first Liouville worked only with expressions   
   > in terms of radicals.  But after reading Abel's work Liouville   
   > realized that roots of polynomials are more general and do not   
   > pose any extra difficulty compared to roots.  So he and followers   
   > consider current definition better, then one insisting of   
   > expressing algebraic quantities in terms of radicals.   
   Precisely because Liouville defined his own term "Elementary functions", his   
   functional class "Elementary fucntions" is different from the general   
   meaning.   
   Am I right?   
      
   Thank you very much for your competent, interesting and fruitful answer.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)