XPost: sci.math   
   From: ivgroups@onlinehome.de   
      
   "Christopher Creutzig" wrote in news:55BF4963.6070803@mathworks.com...   
   >> A symbolic solution: There is an algebraic expression only for this   
   >> solution.   
   > What exactly do you mean by that? “Any x such that x^5-x+1=0” is a   
   > perfectly valid symbolic solution to me, is it to you as well?   
   >   
   > Keyword to look for in the literature (not sure if you find any good   
   > material online): Galois groups, solvable Galois groups.   
   I had expected that this question would come. I wrote (see above): "There is   
   an algebraic expression only for this solution." That should mean: "o n l y   
   for  t h i s  solution". For example: x = A. A an algebraic expression   
   without x. x = A shall be a defining equation for x, only for one   
   representative of the solutions x.   
   Your algebraic equation x^5-x+1=0 is symbolic, but it is not the final   
   solution. But for the radical equations, it is a partial solution. One part   
   of my question was, if each radical equation can be transformed by symbolic   
   (or algebraic?) transformations into a related algebraic equation. I think   
   now, each radical equation can only have algebraic solutions.   
   We know, there are symbolic (= closed form) algebraic solutions (expressible   
   as radical expressions, according to your solvable Galois groups) and   
   non-symbolic algebraic solutions of an algebraic equation. The other part of   
   my original question is, if the non-extraneous (= non-spurious) symbolic   
   solutions of the related algebraic equation are the same as those of the   
   underlying radical equation.   
   I know solvable Galois groups are the key for that. But I guess I cannot   
   find the answer for the radical equations in the literature about Galois   
   groups. Is it possible, that an algebraic solution of a radical equation can   
   be written by an algebraic expression, but the Galois group of the related   
   algebraic equation is not solvable? Does that depend on the method which is   
   used for transforming the radical equation into an algebraic equation? Is   
   there a Galois group of a radical equation, and its solvability decides on   
   the symbolic solutions? Can someone help to answer this naive and simple   
   questions?   
      
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