Package sage :: Package rings :: Package polynomial :: Module polynomial_element_generic :: Class Polynomial_generic_domain

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Class Polynomial_generic_domain

                      object --+                            
                               |                            
structure.sage_object.SageObject --+                        
                                   |                        
           structure.element.Element --+                    
                                       |                    
         structure.element.ModuleElement --+                
                                           |                
               structure.element.RingElement --+            
                                               |            
        structure.element.CommutativeRingElement --+        
                                                   |        
         structure.element.CommutativeAlgebraElement --+    
                                                       |    
                           polynomial_element.Polynomial --+
                                                           |
                          object --+                       |
                                   |                       |
    structure.sage_object.SageObject --+                   |
                                       |                   |
               structure.element.Element --+               |
                                           |               |
             structure.element.ModuleElement --+           |
                                               |           |
                   structure.element.RingElement --+       |
                                                   |       |
            structure.element.CommutativeRingElement --+   |
                                                       |   |
                 structure.element.IntegralDomainElement --+
                                                           |
                                                          Polynomial_generic_domain

Known Subclasses:: Polynomial_generic_field, Polynomial_padic_generic_dense, padics.polynomial_padic_capped_relative_dense.Polynomial_padic_capped_relative_dense

Instance Methods

__init__(self, parent, is_gen=False, construct=False)
x.__init__(...) initializes x; see x.__class__.__doc__ for signature

source code

is_unit(self)
Return True if this polynomial is a unit.

source code

Inherited from polynomial_element.Polynomial: __call__, __copy__, __delitem__, __div__, __eq__, __float__, __floordiv__, __ge__, __getitem__, __gt__, __hash__, __int__, __invert__, __iter__, __le__, __long__, __lshift__, __lt__, __mod__, __ne__, __new__, __nonzero__, __pow__, __rdiv__, __rfloordiv__, __rlshift__, __rmod__, __rpow__, __rrshift__, __rshift__, __setitem__, _fast_float_, _gap_, _gap_init_, _im_gens_, _integer_, _latex_, _magma_, _magma_init_, _pari_, _pari_init_, _rational_, _repr_, args, base_extend, base_ring, change_ring, change_variable_name, coefficients, coeffs, complex_roots, constant_coefficient, degree, denominator, derivative, dict, discriminant, exponents, factor, integral, inverse_mod, inverse_of_unit, is_constant, is_gen, is_irreducible, is_monic, is_nilpotent, is_squarefree, leading_coefficient, list, monic, name, newton_raphson, newton_slopes, norm, ord, padded_list, plot, polynomial, prec, radical, real_roots, resultant, reverse, root_field, roots, shift, square, squarefree_decomposition, subs, substitute, truncate, valuation, variable_name, variables

Inherited from polynomial_element.Polynomial (private): _compile, _derivative, _dict_to_list, _factor_pari_helper, _is_atomic, _lcm, _mpoly_dict_recursive, _mul_fateman, _mul_generic, _mul_karatsuba, _pari_with_name, _pow, _repr, _square_generic, _xgcd

Inherited from structure.element.CommutativeRingElement: divides, mod

Inherited from structure.element.RingElement: __idiv__, __imul__, __mul__, __pos__, __rmul__, __rtruediv__, __truediv__, _div_, _idiv_, _imul_, _mul_, abs, additive_order, is_one, multiplicative_order, order

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

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

Inherited from structure.element.Element: __cmp__, __reduce__, __rxor__, __xor__, _cmp_, _richcmp_, base_base_extend, base_base_extend_canonical_sym, base_extend_canonical, base_extend_canonical_sym, base_extend_recursive, category, is_zero, n, parent

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

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

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

Properties

Inherited from object: __class__

Method Details

init(self, parent, is_gen=False, construct=False)
(Constructor)

x.__init__(...) initializes x; see x.__class__.__doc__ for signature

Overrides: polynomial_element.Polynomial.__init__: (inherited documentation)

is_unit(self)


Return True if this polynomial is a unit.

EXERCISE (Atiyah-McDonald, Ch 1): Let $A[x]$ be a polynomial
ring in one variable.  Then $f=\sum a_i x^i \in A[x]$ is a
unit if and only if $a_0$ is a unit and $a_1,\ldots, a_n$ are
nilpotent.

EXAMPLES:
    sage: R.<z> = PolynomialRing(ZZ, sparse=True)
    sage: (2 + z^3).is_unit()
    False
    sage: f = -1 + 3*z^3; f
    3*z^3 - 1
    sage: f.is_unit()
    False
    sage: R(-3).is_unit()
    False
    sage: R(-1).is_unit()
    True
    sage: R(0).is_unit()
    False

Overrides: polynomial_element.Polynomial.is_unit