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Number Fields

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# Examples of Number Fields
#
# Quadratic Extensions
L=CyclotomicField(3); L
z3=L.gen(); z3
z3^2+z3+1
Cyclotomic Field of order 3 and degree 2 zeta3 0 
L1.<a> = NumberField(x^2 + x + 1); 
L1
a
Number Field in a with defining polynomial x^2 + x + 1 a 
a^2+a+1
z3^2+z3+1
if z3==a: 
    print "z3=a both are zeta3"
else:
    print "different implementations ..."
0 0 different implementations ... 
L.absolute_discriminant()
L1.absolute_discriminant()
-3 -3 
a.trace()
a.norm()
-1 1 
z3.trace()
z3.norm()
-1 1 
# ******************************************************************************************************
# Define the linear automorphism Ma(x)=ax in L

# First some Linear Algebra basics with SAGE
# Constructing a linear transformation
A = matrix(QQ, [[-1, 2, 3], [4, 2, 0]])
phi = linear_transformation(A)
phi
Vector space morphism represented by the matrix: [-1 2 3] [ 4 2 0] Domain: Vector space of dimension 2 over Rational Field Codomain: Vector space of dimension 3 over Rational Field 
phi([2, -3]) # transformations act from right on vectors (at left).
(-14, -2, 6) 
# Different fields as coefficients
F = Integers(13); F
D = F^3; D
C = F^2; C
x, y, z = var('x y z')
f(x, y, z) = [2*x + 3*y + 5*z, x + z]; f
rho = linear_transformation(D, C, f); rho
Ring of integers modulo 13 Vector space of dimension 3 over Ring of integers modulo 13 Vector space of dimension 2 over Ring of integers modulo 13 (x, y, z) |--> (2*x + 3*y + 5*z, x + z) Vector space morphism represented by the matrix: [2 1] [3 0] [5 1] Domain: Vector space of dimension 3 over Ring of integers modulo 13 Codomain: Vector space of dimension 2 over Ring of integers modulo 13 
f(1, 2, 3)
rho([1, 2, 3])
(23, 4) (10, 4) 
# Define M2(x)=2x over F7


# Check the properties of the Discriminant

#