XPost: sci.math   
   From: ivgroups@onlinehome.de   
      
   Hallo.   
      
   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)   
   We know, the defining algebraic equation of some algebraic functions has a   
   solution, but the Galois group of the eqaution is not solvable. Therefore   
   there are algebraic functions which cannot be explicitly described by an   
   algebraic expression.   
   2a) Is there a name for this kind of algebraic functions?   
   2b) Is there a method to construct or describe this kind of algebraic   
   functions only by symbolic methods, e.g. by closed-form expressions?   
      
   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?   
   3b) Has this incorrect definition of the class of elementary functions   
   somewhere any consequences?   
      
   Thanks.   
      
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