#TheoryIndStudy

763 days ago by hwsmith

R.<x> = PolynomialRing(QQ) 
       
K.<a> = NumberField(x^2-3);K 
        
Number Field in a with defining polynomial x^2 - 3
Number Field in a with defining polynomial x^2 - 3
L.<b> = K.extension(x^3-5);L 
        
Number Field in b with defining polynomial x^3 - 5 over its base field
Number Field in b with defining polynomial x^3 - 5 over its base field
L.absolute_polynomial() 
        
x^6 - 9*x^4 - 10*x^3 + 27*x^2 - 90*x - 2
x^6 - 9*x^4 - 10*x^3 + 27*x^2 - 90*x - 2
M.<c> = L.absolute_field();M 
        
Number Field in c with defining polynomial x^6 - 9*x^4 - 10*x^3 + 27*x^2
- 90*x - 2
Number Field in c with defining polynomial x^6 - 9*x^4 - 10*x^3 + 27*x^2 - 90*x - 2
N = M.optimized_representation()[0];N 
        
Number Field in c1 with defining polynomial x^6 - 24*x^4 + 192*x^2 - 507
Number Field in c1 with defining polynomial x^6 - 24*x^4 + 192*x^2 - 507
M.is_isomorphic(N) 
        
True
True
S.<y> = M[];S 
        
Univariate Polynomial Ring in y over Number Field in c with defining
polynomial x^6 - 9*x^4 - 10*x^3 + 27*x^2 - 90*x - 2
Univariate Polynomial Ring in y over Number Field in c with defining polynomial x^6 - 9*x^4 - 10*x^3 + 27*x^2 - 90*x - 2
factor(y^2-3) 
        
(y - 8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 - 335/267*c +
455/267) * (y + 8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 +
335/267*c - 455/267)
(y - 8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 - 335/267*c + 455/267) * (y + 8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 + 335/267*c - 455/267)
T.<z> = N[] 
       
factor(z^2-3) 
        
(z - 2/13*c1^5 + 35/13*c1^3 - 150/13*c1) * (z + 2/13*c1^5 - 35/13*c1^3 +
150/13*c1)
(z - 2/13*c1^5 + 35/13*c1^3 - 150/13*c1) * (z + 2/13*c1^5 - 35/13*c1^3 + 150/13*c1)
alpha = - 8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 - 335/267*c +455/267;alpha 
        
-8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 - 335/267*c + 455/267
-8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 - 335/267*c + 455/267
factor(y^3-5) 
        
(y + 8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 + 68/267*c -
455/267) * (y^2 + (-8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 -
68/267*c + 455/267)*y - 10/267*c^5 + 16/267*c^4 + 100/267*c^3 +
29/267*c^2 - 530/267*c + 769/267)
(y + 8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 + 68/267*c - 455/267) * (y^2 + (-8/267*c^5 - 5/267*c^4 + 80/267*c^3 + 130/267*c^2 - 68/267*c + 455/267)*y - 10/267*c^5 + 16/267*c^4 + 100/267*c^3 + 29/267*c^2 - 530/267*c + 769/267)
beta = 8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 + 68/267*c -455/267;beta 
        
8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 + 68/267*c - 455/267
8/267*c^5 + 5/267*c^4 - 80/267*c^3 - 130/267*c^2 + 68/267*c - 455/267
elt = alpha+beta;elt 
        
-c
-c
elt.minpoly() 
        
x^6 - 9*x^4 + 10*x^3 + 27*x^2 + 90*x - 2
x^6 - 9*x^4 + 10*x^3 + 27*x^2 + 90*x - 2
alpha.minpoly() 
        
x^2 - 3
x^2 - 3