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    me                     @   sn   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 e
G dd	 d	eeZe ZZd
S )z.Implementation of :class:`FiniteField` class.     )Field)ModularIntegerFactory)SimpleDomain)CoercionFailed)public)SymPyIntegerc                   @   s   e Zd ZdZdZdZd ZZdZdZ	dZ
dZdZd*ddZdd	 Zd
d Zdd Zdd Zdd Zdd Zdd Zd+ddZd,ddZd-ddZd.ddZd/ddZd0d d!Zd1d"d#Zd2d$d%Zd3d&d'Zd(d) ZdS )4FiniteFielda	  Finite field of prime order :ref:`GF(p)`

    A :ref:`GF(p)` domain represents a `finite field`_ `\mathbb{F}_p` of prime
    order as :py:class:`~.Domain` in the domain system (see
    :ref:`polys-domainsintro`).

    A :py:class:`~.Poly` created from an expression with integer
    coefficients will have the domain :ref:`ZZ`. However, if the ``modulus=p``
    option is given then the domain will be a finite field instead.

    >>> from sympy import Poly, Symbol
    >>> x = Symbol('x')
    >>> p = Poly(x**2 + 1)
    >>> p
    Poly(x**2 + 1, x, domain='ZZ')
    >>> p.domain
    ZZ
    >>> p2 = Poly(x**2 + 1, modulus=2)
    >>> p2
    Poly(x**2 + 1, x, modulus=2)
    >>> p2.domain
    GF(2)

    It is possible to factorise a polynomial over :ref:`GF(p)` using the
    modulus argument to :py:func:`~.factor` or by specifying the domain
    explicitly. The domain can also be given as a string.

    >>> from sympy import factor, GF
    >>> factor(x**2 + 1)
    x**2 + 1
    >>> factor(x**2 + 1, modulus=2)
    (x + 1)**2
    >>> factor(x**2 + 1, domain=GF(2))
    (x + 1)**2
    >>> factor(x**2 + 1, domain='GF(2)')
    (x + 1)**2

    It is also possible to use :ref:`GF(p)` with the :py:func:`~.cancel`
    and :py:func:`~.gcd` functions.

    >>> from sympy import cancel, gcd
    >>> cancel((x**2 + 1)/(x + 1))
    (x**2 + 1)/(x + 1)
    >>> cancel((x**2 + 1)/(x + 1), domain=GF(2))
    x + 1
    >>> gcd(x**2 + 1, x + 1)
    1
    >>> gcd(x**2 + 1, x + 1, domain=GF(2))
    x + 1

    When using the domain directly :ref:`GF(p)` can be used as a constructor
    to create instances which then support the operations ``+,-,*,**,/``

    >>> from sympy import GF
    >>> K = GF(5)
    >>> K
    GF(5)
    >>> x = K(3)
    >>> y = K(2)
    >>> x
    3 mod 5
    >>> y
    2 mod 5
    >>> x * y
    1 mod 5
    >>> x / y
    4 mod 5

    Notes
    =====

    It is also possible to create a :ref:`GF(p)` domain of **non-prime**
    order but the resulting ring is **not** a field: it is just the ring of
    the integers modulo ``n``.

    >>> K = GF(9)
    >>> z = K(3)
    >>> z
    3 mod 9
    >>> z**2
    0 mod 9

    It would be good to have a proper implementation of prime power fields
    (``GF(p**n)``) but these are not yet implemented in SymPY.

    .. _finite field: https://en.wikipedia.org/wiki/Finite_field
    FFTFNc                 C   s\   ddl m} |}|dkr$td| t|||| | _| d| _| d| _|| _|| _d S )Nr   )ZZz*modulus must be a positive integer, got %s   )	Zsympy.polys.domainsr
   
ValueErrorr   dtypeZzeroZonedommod)selfr   Z	symmetricr
   r    r   C/tmp/pip-unpacked-wheel-rdz2gdd2/sympy/polys/domains/finitefield.py__init__r   s    zFiniteField.__init__c                 C   s
   d| j  S )NzGF(%s)r   r   r   r   r   __str__   s    zFiniteField.__str__c                 C   s   t | jj| j| j| jfS )N)hash	__class____name__r   r   r   r   r   r   r   __hash__   s    zFiniteField.__hash__c                 C   s"   t |to | j|jko | j|jkS )z0Returns ``True`` if two domains are equivalent. )
isinstancer   r   r   )r   otherr   r   r   __eq__   s
    


zFiniteField.__eq__c                 C   s   | j S )z*Return the characteristic of this domain. r   r   r   r   r   characteristic   s    zFiniteField.characteristicc                 C   s   | S )z*Returns a field associated with ``self``. r   r   r   r   r   	get_field   s    zFiniteField.get_fieldc                 C   s   t t|S )z!Convert ``a`` to a SymPy object. )r   intr   ar   r   r   to_sympy   s    zFiniteField.to_sympyc                 C   sT   |j r| | jt|S |jrDt||krD| | jt|S td| dS )z0Convert SymPy's Integer to SymPy's ``Integer``. zexpected an integer, got %sN)Z
is_Integerr   r   r    Zis_Floatr   r!   r   r   r   
from_sympy   s
    zFiniteField.from_sympyc                 C   s   |  | j|j|jS z.Convert ``ModularInteger(int)`` to ``dtype``. )r   r   from_ZZvalK1r"   K0r   r   r   from_FF   s    zFiniteField.from_FFc                 C   s   |  | j|j|jS r%   )r   r   from_ZZ_pythonr'   r(   r   r   r   from_FF_python   s    zFiniteField.from_FF_pythonc                 C   s   |  | j||S z'Convert Python's ``int`` to ``dtype``. r   r   r,   r(   r   r   r   r&      s    zFiniteField.from_ZZc                 C   s   |  | j||S r.   r/   r(   r   r   r   r,      s    zFiniteField.from_ZZ_pythonc                 C   s   |j dkr| |jS dS z,Convert Python's ``Fraction`` to ``dtype``. r   Ndenominatorr,   	numeratorr(   r   r   r   from_QQ   s    
zFiniteField.from_QQc                 C   s   |j dkr| |jS dS r0   r1   r(   r   r   r   from_QQ_python   s    
zFiniteField.from_QQ_pythonc                 C   s   |  | j|j|jS )z.Convert ``ModularInteger(mpz)`` to ``dtype``. )r   r   from_ZZ_gmpyr'   r(   r   r   r   from_FF_gmpy   s    zFiniteField.from_FF_gmpyc                 C   s   |  | j||S )z%Convert GMPY's ``mpz`` to ``dtype``. )r   r   r6   r(   r   r   r   r6      s    zFiniteField.from_ZZ_gmpyc                 C   s   |j dkr| |jS dS )z%Convert GMPY's ``mpq`` to ``dtype``. r   N)r2   r6   r3   r(   r   r   r   from_QQ_gmpy   s    
zFiniteField.from_QQ_gmpyc                 C   s,   | |\}}|dkr(| | j|S dS )z'Convert mpmath's ``mpf`` to ``dtype``. r   N)Zto_rationalr   r   )r)   r"   r*   pqr   r   r   from_RealField   s    zFiniteField.from_RealField)T)N)N)N)N)N)N)N)N)N)r   
__module____qualname____doc__repaliasZis_FiniteFieldZis_FFZis_NumericalZhas_assoc_RingZhas_assoc_Fieldr   r   r   r   r   r   r   r   r#   r$   r+   r-   r&   r,   r4   r5   r7   r6   r8   r;   r   r   r   r   r      s6   X
	
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

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r   N)r>   Zsympy.polys.domains.fieldr   Z"sympy.polys.domains.modularintegerr   Z sympy.polys.domains.simpledomainr   Zsympy.polys.polyerrorsr   Zsympy.utilitiesr   Zsympy.polys.domains.groundtypesr   r   r	   ZGFr   r   r   r   <module>   s    B