<s>[INST] Splitting field in math software. Can anyone help me to find a method for calculating the splitting field for a polynomial over a function field?
I think this feature is not currently supported..
 [/INST] You can find the splitting field with the Isprime and PrimaryDecomposition function
     P<x,y,z>  := PolynomialRing(Rationals(),3); 
     I := ideal<P| x^2+y^3+y, z^3+z*(x^3+y)+1>;
     IsPrime(I);  // true
     R := quo<P|I>; x:= R.1; y := R.2; z := R.3; Q<w> := PolynomialRing(R,1); 
     f := w^3+w*(x^3+y)+1; g:= f div (w-z);  g  


     P<x,y,z,w>  := PolynomialRing(Rationals(),4);
     I := ideal<P|x^2+y^3+y, z^3+z*(x^3+y)+1,w^2 + z*w + y^9*z + 3*y^7*z + 3*y^5*z + y^3*z + y^2*z + 2*y*z^3 + 2*y + z^5 +  2*z^2>;
     IsPrime(I); // true

     R := quo<P|I>; x:= R.1; y := R.2; z := R.3; w := R.4; Q<u> := 
     PolynomialRing(R,1); 
     f := u^2 + z*u + y^9*z + 3*y^7*z + 3*y^5*z + y^3*z + y^2*z + 2*y*z^3 + 2*y + z^5 +
2*z^2; f div (u-w)         //  = u + z + w

thus $$T^3+T(x^3+y)+1 = (T-z)(T-w)(T-(u+z+w)) \\\in \Bbb{Q}[x,y,z,w]/J [T]$$
where $$J=(x^2+y^3+y , \qquad \qquad \qquad \qquad \qquad \qquad \\ \qquad \qquad  z^3+z(x^3+y)+1, \qquad \qquad \qquad\\ \qquad \qquad w^2 + z w + y^9 z + 3 y^7 z + 3 y^5 z + y^3 z + y^2 z + 2 y z^3 + 2 y + z^5 +  2 z^2)  $$
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