Package sage :: Package algebras :: Module quaternion_order_element :: Class QuaternionOrderElement

Class QuaternionOrderElement

                      object --+                        
                               |                        
structure.sage_object.SageObject --+                    
                                   |                    
           structure.element.Element --+                
                                       |                
         structure.element.ModuleElement --+            
                                           |            
               structure.element.RingElement --+        
                                               |        
                structure.element.AlgebraElement --+    
                                                   |    
           algebra_order_element.AlgebraOrderElement --+
                                                       |
                                                      QuaternionOrderElement

Known Subclasses:: quaternion_order_ideal_element.QuaternionOrderIdealElement

Instance Methods

[hide private]

__init__(self, A, x, check=True)
Create the element x of the quaternion order A.

source code

__eq__(self, x)
x==y

source code

conjugate(self)

source code

reduced_trace(self)
Return the reduced trace of this element.

source code

reduced_norm(self)

source code

charpoly(self, var)

source code

characteristic_polynomial(self, var)

source code

minpoly(self, var)

source code

minimal_polynomial(self, var)

source code

vector(self)

source code

Inherited from algebra_order_element.AlgebraOrderElement: __add__, __mul__, __neg__, __radd__, __repr__, __sub__, ambient_algebra_element, norm, trace

Inherited from structure.element.AlgebraElement: __new__

Inherited from structure.element.RingElement: __div__, __idiv__, __imul__, __invert__, __pos__, __pow__, __rdiv__, __rmul__, __rpow__, __rtruediv__, __truediv__, _div_, _idiv_, _imul_, _mul_, abs, additive_order, is_nilpotent, is_one, is_unit, multiplicative_order, order

Inherited from structure.element.ModuleElement: __iadd__, __isub__, __rsub__, _add_, _iadd_, _ilmul_, _isub_, _lmul_, _neg_, _rmul_, _sub_

Inherited from structure.element.ModuleElement (private): _lmul_nonscalar, _rmul_nonscalar

Inherited from structure.element.Element: __cmp__, __ge__, __gt__, __hash__, __le__, __lt__, __ne__, __nonzero__, __reduce__, __rxor__, __xor__, _cmp_, _im_gens_, _repr_, _richcmp_, base_base_extend, base_base_extend_canonical_sym, base_extend, base_extend_canonical, base_extend_canonical_sym, base_extend_recursive, base_ring, category, is_zero, n, parent, subs, substitute

Inherited from structure.element.Element (private): _coeff_repr, _is_atomic, _latex_coeff_repr, _make_new_with_parent_c, _set_parent

Inherited from structure.sage_object.SageObject: _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _magma_init_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from object: __delattr__, __getattribute__, __reduce_ex__, __setattr__, __str__

Properties

[hide private]

Inherited from object: __class__

Method Details

[hide private]

init(self, A, x, check=True)
(Constructor)

source code


Create the element x of the quaternion order A.

Overrides: algebra_order_element.AlgebraOrderElement.__init__

eq(self, x)
(Equality operator)

source code

x==y

Overrides: structure.element.Element.__eq__: (inherited documentation)

reduced_trace(self)

source code


Return the reduced trace of this element.

\note{In a quaternion algebra $A$, every element $x$ is
quadratic over the center, thus $x^2 = \Tr(x)*x - \Nr(x)$, so
we solve for a linear relation $(1,-\Tr(x),\Nr(x))$ among
$[x^2, x, 1]$ for the reduced trace of $x$.}

Class QuaternionOrderElement

__init__(self, A, x, check=True) (Constructor)

__eq__(self, x) (Equality operator)

reduced_trace(self)

charpoly(self, var)

characteristic_polynomial(self, var)

minpoly(self, var)

minimal_polynomial(self, var)

init(self, A, x, check=True)
(Constructor)

eq(self, x)
(Equality operator)