The Field of Algebraic Numbers

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu

4/4/2011; last edit:10/24/2011

Warm-up: The Additive Inverse of an Algebraic Number is Algebraic.

P[x_]=x^7-4x^6+2x^5-2x^3+3x^2-x+3

Sort[NSolve[P[x]==0][[All,1,2]],If[Re[#1]==Re[#2],Im[#1]<Im[#2],Re[#1]<Re[#2]]&]

Q[x_]=-x^7-4x^6-2x^5+2x^3+3x^2+x+3

Sort[NSolve[Q[x]==0][[All,1,2]],If[Re[#1]==Re[#2],Im[#1]>Im[#2],Re[#1]>Re[#2]]&]

The Resultant

Resultant[(x-a)(x-b),(x-c)(x-d)(x-e),x]

P[x_]=x^4-5
Q[x_]=x^4-2x^2-5

Solve[P[x]==0]

Solve[Q[x]==0]

Resultant[P[x],Q[x],x]

The Sum of Two Algebraic Numbers is Algebraic.

Clear[R]
R[z_]=Resultant[P[x],Q[z-x],x]

NSolve[R[z]==0]

- 5.^.25+-I (Sqrt[(Sqrt[6])-1])

The Product of Two Algebraic Numbers is Algebraic.

Resultant[P[x],x^2Q[t/x],x]

Solve[%==0]

N[%]

(- 5.^.25)(-I (Sqrt[(Sqrt[6])-1]))

The Multiplicative Inverse of an Algebraic Numbers is Algebraic.

Solve[y^4 Q[1/y]==0]

N[%]

1/(-I (Sqrt[(Sqrt[6.])-1]))