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"""Implementation of :class:`FiniteField` class. """
from sympy.polys.domains.field import Field
from sympy.polys.domains.modularinteger import ModularIntegerFactory
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
from sympy.polys.domains.groundtypes import SymPyInteger
@public
class FiniteField(Field, SimpleDomain):
r"""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
"""
rep = 'FF'
alias = 'FF'
is_FiniteField = is_FF = True
is_Numerical = True
has_assoc_Ring = False
has_assoc_Field = True
dom = None
mod = None
def __init__(self, mod, symmetric=True):
from sympy.polys.domains import ZZ
dom = ZZ
if mod <= 0:
raise ValueError('modulus must be a positive integer, got %s' % mod)
self.dtype = ModularIntegerFactory(mod, dom, symmetric, self)
self.zero = self.dtype(0)
self.one = self.dtype(1)
self.dom = dom
self.mod = mod
def __str__(self):
return 'GF(%s)' % self.mod
def __hash__(self):
return hash((self.__class__.__name__, self.dtype, self.mod, self.dom))
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, FiniteField) and \
self.mod == other.mod and self.dom == other.dom
def characteristic(self):
"""Return the characteristic of this domain. """
return self.mod
def get_field(self):
"""Returns a field associated with ``self``. """
return self
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return SymPyInteger(int(a))
def from_sympy(self, a):
"""Convert SymPy's Integer to SymPy's ``Integer``. """
if a.is_Integer:
return self.dtype(self.dom.dtype(int(a)))
elif a.is_Float and int(a) == a:
return self.dtype(self.dom.dtype(int(a)))
else:
raise CoercionFailed("expected an integer, got %s" % a)
def from_FF(K1, a, K0=None):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ(a.val, K0.dom))
def from_FF_python(K1, a, K0=None):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ_python(a.val, K0.dom))
def from_ZZ(K1, a, K0=None):
"""Convert Python's ``int`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ_python(a, K0))
def from_ZZ_python(K1, a, K0=None):
"""Convert Python's ``int`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ_python(a, K0))
def from_QQ(K1, a, K0=None):
"""Convert Python's ``Fraction`` to ``dtype``. """
if a.denominator == 1:
return K1.from_ZZ_python(a.numerator)
def from_QQ_python(K1, a, K0=None):
"""Convert Python's ``Fraction`` to ``dtype``. """
if a.denominator == 1:
return K1.from_ZZ_python(a.numerator)
def from_FF_gmpy(K1, a, K0=None):
"""Convert ``ModularInteger(mpz)`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ_gmpy(a.val, K0.dom))
def from_ZZ_gmpy(K1, a, K0=None):
"""Convert GMPY's ``mpz`` to ``dtype``. """
return K1.dtype(K1.dom.from_ZZ_gmpy(a, K0))
def from_QQ_gmpy(K1, a, K0=None):
"""Convert GMPY's ``mpq`` to ``dtype``. """
if a.denominator == 1:
return K1.from_ZZ_gmpy(a.numerator)
def from_RealField(K1, a, K0):
"""Convert mpmath's ``mpf`` to ``dtype``. """
p, q = K0.to_rational(a)
if q == 1:
return K1.dtype(K1.dom.dtype(p))
FF = GF = FiniteField
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