
    sgE                          d Z ddlmZ ddlmZ ddlmZ ddlmZ ddl	m
Z
 ddlmZmZmZ dd	lmZ d
 ZddZd Ze
dd       Zy)z,Computing integral bases for number fields.     )Poly)AlgebraicField)ZZ)QQ)public   )ModuleEndomorphismModuleHomomorphism
PowerBasis) extract_fundamental_discriminantc                    | j                   }t        | |      }|j                         \  }}|dk(  sJ t        d||      }|D ]
  \  }}||z  } ||z  }	t        |t              }
t        |	t              }|
|z  | z
  |z  }t        ||      }|}||	fD ]  }|j	                  |      } ||z  }|j                         }||fS )zz
    Apply the "Dedekind criterion" to test whether the order needs to be
    enlarged relative to a given prime *p*.
    modulusr   domain)genr   factor_listr   gcddegree)TpxT_barlcflg_barti_bar_h_barghff_barZ_barbU_barms                     Q/var/www/html/venv/lib/python3.12/site-packages/sympy/polys/numberfields/basis.py_apply_Dedekind_criterionr)      s    
 	
AAE FB7N7Aq!E 	UNEU2AU2A	
QqAAEEU^ 		!UNEA!8O    Nc                     | j                   }||k  r|z  |k  rt        | fd      }|j                  |      S )a  
    Compute the nilradical mod *p* for a given order *H*, and prime *p*.

    Explanation
    ===========

    This is the ideal $I$ in $H/pH$ consisting of all elements some positive
    power of which is zero in this quotient ring, i.e. is a multiple of *p*.

    Parameters
    ==========

    H : :py:class:`~.Submodule`
        The given order.
    p : int
        The rational prime.
    q : int, optional
        If known, the smallest power of *p* that is $>=$ the dimension of *H*.
        If not provided, we compute it here.

    Returns
    =======

    :py:class:`~.Module` representing the nilradical mod *p* in *H*.

    References
    ==========

    .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory*.
    (See Lemma 6.1.6.)

    c                     | z  S N )r   qs    r(   <lambda>z"nilradical_mod_p.<locals>.<lambda>K   s    !Q$ r*   r   )nr	   kernel)Hr   r/   r1   phis     `  r(   nilradical_mod_pr5   %   sO    B 	
Ay!eFA !e
Q
/C::a:  r*   c                   
 t        | ||      }| j                  j                  | j                  |j                  z  | j                        }||| z  z   }|j                         
t        | 

fd      }|j                  |      }| j                  j                  | j                  |j                  z  | j                  |z        }|| z   }	|	|fS )zD
    Perform the second enlargement in the Round Two algorithm.
    )r/   )denomc                 &    j                  |       S r-   )inner_endomorphism)r   Es    r(   r0   z%_second_enlargement.<locals>.<lambda>W   s    Q-A-A!-D r*   r   )r5   parentsubmodule_from_matrixmatrixr7   endomorphism_ringr
   r2   )r3   r   r/   IpBCr4   gammaGH1r:   s             @r(   _second_enlargementrE   O   s     
!Q!	$B	&&qxx"))';177&KA	AaCA	A
Q#D
ECJJqJ!E	&&qxx%,,'>aggPQk&RA	
QBr6Mr*   c                    d}t        | t              r| | j                  j                         } }| j                  r$| j
                  r| j                  t        t        fvrt        d      | j                         \  } }| j                         }| j                         }t        j                  t        |            }t        |      \  }}t!        |xs |       }|j#                         }	d}
|r|j%                         \  }}t'        | |      \  }}|dk(  r*|j)                  t+        |t                    }|	j-                  ||z  |	z  |      }	||k  ri|}||k  r||z  }||k  rt/        |	||      \  }}
||	k7  r|}	t/        |	||      \  }}
||	k7  r|r|
t        |t0              r|
|<   |	}d|_        d|_        ||j6                  j9                         dz  z  |j:                  d|z  z  z  }||fS )a  
    Zassenhaus's "Round 2" algorithm.

    Explanation
    ===========

    Carry out Zassenhaus's "Round 2" algorithm on an irreducible polynomial
    *T* over :ref:`ZZ` or :ref:`QQ`. This computes an integral basis and the
    discriminant for the field $K = \mathbb{Q}[x]/(T(x))$.

    Alternatively, you may pass an :py:class:`~.AlgebraicField` instance, in
    place of the polynomial *T*, in which case the algorithm is applied to the
    minimal polynomial for the field's primitive element.

    Ordinarily this function need not be called directly, as one can instead
    access the :py:meth:`~.AlgebraicField.maximal_order`,
    :py:meth:`~.AlgebraicField.integral_basis`, and
    :py:meth:`~.AlgebraicField.discriminant` methods of an
    :py:class:`~.AlgebraicField`.

    Examples
    ========

    Working through an AlgebraicField:

    >>> from sympy import Poly, QQ
    >>> from sympy.abc import x
    >>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
    >>> K = QQ.alg_field_from_poly(T, "theta")
    >>> print(K.maximal_order())
    Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2
    >>> print(K.discriminant())
    -503
    >>> print(K.integral_basis(fmt='sympy'))
    [1, theta, theta/2 + theta**2/2]

    Calling directly:

    >>> from sympy import Poly
    >>> from sympy.abc import x
    >>> from sympy.polys.numberfields.basis import round_two
    >>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
    >>> print(round_two(T))
    (Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2, -503)

    The nilradicals mod $p$ that are sometimes computed during the Round Two
    algorithm may be useful in further calculations. Pass a dictionary under
    `radicals` to receive these:

    >>> T = Poly(x**3 + 3*x**2 + 5)
    >>> rad = {}
    >>> ZK, dK = round_two(T, radicals=rad)
    >>> print(rad)
    {3: Submodule[[-1, 1, 0], [-1, 0, 1]]}

    Parameters
    ==========

    T : :py:class:`~.Poly`, :py:class:`~.AlgebraicField`
        Either (1) the irreducible polynomial over :ref:`ZZ` or :ref:`QQ`
        defining the number field, or (2) an :py:class:`~.AlgebraicField`
        representing the number field itself.

    radicals : dict, optional
        This is a way for any $p$-radicals (if computed) to be returned by
        reference. If desired, pass an empty dictionary. If the algorithm
        reaches the point where it computes the nilradical mod $p$ of the ring
        of integers $Z_K$, then an $\mathbb{F}_p$-basis for this ideal will be
        stored in this dictionary under the key ``p``. This can be useful for
        other algorithms, such as prime decomposition.

    Returns
    =======

    Pair ``(ZK, dK)``, where:

        ``ZK`` is a :py:class:`~sympy.polys.numberfields.modules.Submodule`
        representing the maximal order.

        ``dK`` is the discriminant of the field $K = \mathbb{Q}[x]/(T(x))$.

    See Also
    ========

    .AlgebraicField.maximal_order
    .AlgebraicField.integral_basis
    .AlgebraicField.discriminant

    References
    ==========

    .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*

    NzDRound 2 requires an irreducible univariate polynomial over ZZ or QQ.r   r   )hnf_modulusT   )
isinstancer   extminpoly_of_elementis_univariateis_irreducibler   r   r   
ValueError)make_monic_over_integers_by_scaling_rootsr   discriminant
from_sympyabsr   r   whole_submodulepopitemr)   element_from_polyr   addrE   dict_starts_with_unity_is_sq_maxrank_HNFr=   detr7   )r   radicalsKr   r1   D	D_modulusFZthetar3   nilradr   er&   r'   Ur/   rD   ZKdKs                       r(   	round_tworf   ^   s   @ 	A!^$!%%**,188B8#_``668DAq	
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 EE!q&1*)E46 !eFA !e(Aq1
FAgA,Q15JB Ag- @ j48	
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"rxxAE':	:Br6Mr*   r-   )__doc__sympy.polys.polytoolsr   "sympy.polys.domains.algebraicfieldr   sympy.polys.domains.integerringr   !sympy.polys.domains.rationalfieldr   sympy.utilities.decoratorr   modulesr	   r
   r   	utilitiesr   r)   r5   rE   rf   r.   r*   r(   <module>ro      sF    2 & = . 0 , G G 72'!T W Wr*   