Clear[x]

Solve[1-x+(x^5)==0,x]

Root[f,k]
represents the exact k(^th) root of the polynomial equation f[x]==0.

Solve[1-x+(x^5)==0,x]//N

Root[1-x+(x^5),1]//N

({{ToNumberField[a,θ]   express the algebraic number a in the number field generated by θ                               }, {ToNumberField[{(Subscript[a, 1]),(Subscript[a, 2]),…},θ]   express the (Subscript[a, i])in the field generated by θ                                                                  }, {ToNumberField[{(Subscript[a, 1]),(Subscript[a, 2]),…}]   express the (Subscript[a, i])in a common extension field generated by a                            
                                                             single  algebraic number}})

ToNumberField[{(Sqrt[2]),(Sqrt[3])}]

AlgebraicNumber[θ,{(Subscript[c, 0]),(Subscript[c, 1]),…,(Subscript[c, n])}]  represent the algebraic number (Subscript[c, 0])+(Subscript[c, 1])θ +…+(Subscript[c, n]) (θ^n) in ℚ[θ]

ra=Root[1-10 ((#1)^2)+((#1)^4)&,4]//N;
(Sqrt[2])==-(9/2) ra+(1/2)ra^3

AlgebraicNumber automatically makes the generator of the extension an algebraic integer and the coefficient list equal in length to the degree of the extension.

A=AlgebraicNumber[Root[2 ((#)^4)-3 #+2&,1],{1,2,3,4,5,6}]

A//N

r=Root[2 ((#)^4)-3 #+2&,1];

F[x_]=1+2 x+3 x^2+4 x^3+5 x^4+6 x^5;

F[r]//N

b=AlgebraicNumber[Root[16-12 #1+((#1)^4)&,1],{-4,(7/4),3,(1/2)}]//N

Das Beispiel von vorher:

ToNumberField[{(Sqrt[2]),(Sqrt[3])}]

ToNumberField[{(Sqrt[2]),(Sqrt[3])},Root[1-10 ((#1)^2)+((#1)^4)&,1]]

z=2(Sqrt[2]);
y=3(Sqrt[3]);

z/2+y/3+1/(z/2-y/3)//N

ToNumberField[(Sqrt[2])+(Sqrt[3])+(1/((Sqrt[2])-(Sqrt[3])))]

RootReduce transforms AlgebraicNumber objects to Root objects.

A

RootReduce[A]

Arithmetic within a fixed finite extension of rationals is much faster than arithmetic within the field of all complex algebraic numbers.

Clear[x,y,z]

{a,b,c}={I,(Sqrt[2]),Root[((#)^3)-2 #+3&,1]};
f=((-2 y z (7+x-y+(z^2))+(6+(x^2)+2 y) (-11+x y+(z^2)))/(2 y z (-4-x+3 y z)-(6+(x^2)+2 y) (2-2 x+(z^3))));

RootReduce[h=f/.{x->a,y->b,z->c}]//Timing

h//N

({aa,bb,cc}=ToNumberField[{a,b,c}])//Timing

(d=((-2 y z (7+x-y+(z^2))+(6+(x^2)+2 y) (-11+x y+(z^2)))/(2 y z (-4-x+3 y z)-(6+(x^2)+2 y) (2-2 x+(z^3))))/.{x->aa,y->bb,z->cc})//Timing

d//N

MinimalPolynomial[a]    give a pure function representation of the minimal polynomial over the integers of the algebraic number a
MinimalPolynomial[a,x]
give the minimal polynomial of the algebraic number a as a polynomial in x

MinimalPolynomial[(Sqrt[2])+(Sqrt[3])]

MinimalPolynomial[(Sqrt[2])+(Sqrt[3]),x]

AlgebraicIntegerQ[(1/2) (1+(Sqrt[5]))]

MinimalPolynomial[(1/2) (1+(Sqrt[5]))]

AlgebraicIntegerQ[(1/4) (1+(Sqrt[5]))]

MinimalPolynomial[(1/4) (1+(Sqrt[5]))]