clicliclic@freenet.de  wrote:   
   >   
   > Hello,   
   >   
   > can this trivariate polynomial:   
   >   
   > 256*a^5*c^3 - a^4*(128*b^2*c^2 - 144*b*c - 512*c^4 + 27)   
   >  + a^3*(16*b^4*c - 4*b^3 - 512*b^2*c^3 + 208*b*c^2 + 4*c*(64*c^4 - 9))   
   >  + 2*a^2*(80*b^4*c^2 - 82*b^3*c + b^2*(15 - 64*c^4) - 104*b*c^3 - c^2)   
   >  - 4*a*(4*b^6*c - b^5 - 4*b^4*c^3 - 41*b^3*c^2 + 35*b^2*c   
   >        + 6*b*(6*c^4 - 1) + 9*c^3)   
   >  - 4*b^5*c + b^4 + 4*b^3*c^3 + 30*b^2*c^2 - 24*b*c - 27*c^4 + 4   
   >   
   > be factored over the algebraic extension combining 2^(1/4), 3^(1/8),   
   > and SQRT(SQRT(3) + 1)?   
      
   No.  It is irreducible over integers (any system can check this).   
   Galois theory says that factors over algebraic extensions must   
   be of equal degree.  Since degree with respect to a is 5, and with   
   respect to b is 6 there is no way to factor it into factors of   
   equal degree, so it is irreducible over any algebraic extension   
   of integers.   
      
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