105 lines
2.5 KiB
Python
105 lines
2.5 KiB
Python
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"""Implementation of :class:`Field` class. """
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from sympy.polys.domains.ring import Ring
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from sympy.polys.polyerrors import NotReversible, DomainError
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from sympy.utilities import public
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@public
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class Field(Ring):
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"""Represents a field domain. """
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is_Field = True
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is_PID = True
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def get_ring(self):
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"""Returns a ring associated with ``self``. """
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raise DomainError('there is no ring associated with %s' % self)
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def get_field(self):
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"""Returns a field associated with ``self``. """
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return self
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def exquo(self, a, b):
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"""Exact quotient of ``a`` and ``b``, implies ``__truediv__``. """
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return a / b
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def quo(self, a, b):
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"""Quotient of ``a`` and ``b``, implies ``__truediv__``. """
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return a / b
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def rem(self, a, b):
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"""Remainder of ``a`` and ``b``, implies nothing. """
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return self.zero
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def div(self, a, b):
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"""Division of ``a`` and ``b``, implies ``__truediv__``. """
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return a / b, self.zero
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def gcd(self, a, b):
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"""
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Returns GCD of ``a`` and ``b``.
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This definition of GCD over fields allows to clear denominators
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in `primitive()`.
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Examples
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========
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>>> from sympy.polys.domains import QQ
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>>> from sympy import S, gcd, primitive
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>>> from sympy.abc import x
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>>> QQ.gcd(QQ(2, 3), QQ(4, 9))
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2/9
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>>> gcd(S(2)/3, S(4)/9)
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2/9
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>>> primitive(2*x/3 + S(4)/9)
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(2/9, 3*x + 2)
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"""
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try:
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ring = self.get_ring()
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except DomainError:
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return self.one
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p = ring.gcd(self.numer(a), self.numer(b))
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q = ring.lcm(self.denom(a), self.denom(b))
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return self.convert(p, ring)/q
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def lcm(self, a, b):
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"""
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Returns LCM of ``a`` and ``b``.
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>>> from sympy.polys.domains import QQ
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>>> from sympy import S, lcm
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>>> QQ.lcm(QQ(2, 3), QQ(4, 9))
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4/3
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>>> lcm(S(2)/3, S(4)/9)
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4/3
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"""
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try:
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ring = self.get_ring()
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except DomainError:
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return a*b
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p = ring.lcm(self.numer(a), self.numer(b))
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q = ring.gcd(self.denom(a), self.denom(b))
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return self.convert(p, ring)/q
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def revert(self, a):
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"""Returns ``a**(-1)`` if possible. """
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if a:
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return 1/a
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else:
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raise NotReversible('zero is not reversible')
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def is_unit(self, a):
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"""Return true if ``a`` is a invertible"""
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return bool(a)
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