Domain
>>> x, y = symbols('x, y')
>>> p = Poly(x**2 + y)
>>> p
Poly(x**2 + y, x, y, domain='ZZ')
>>> p.domain
ZZ
>>> isinstance(p.domain, Domain)
True
>>> Poly(x**2 + y/2)
Poly(x**2 + 1/2*y, x, y, domain='QQ')

The domains can be used directly in which case the domain object e.g.
(:ref:`ZZ` or :ref:`QQ`) can be used as a constructor for elements of
``dtype``.

>>> from sympy import ZZ, QQ
>>> ZZ(2)
2
>>> ZZ.dtype  # doctest: +SKIP
<class 'int'>
>>> type(ZZ(2))  # doctest: +SKIP
<class 'int'>
>>> QQ(1, 2)
1/2
>>> type(QQ(1, 2))  # doctest: +SKIP
<class 'sympy.polys.domains.pythonrational.PythonRational'>

The corresponding domain elements can be used with the arithmetic
operations ``+,-,*,**`` and depending on the domain some combination of
``/,//,%`` might be usable. For example in :ref:`ZZ` both ``//`` (floor
division) and ``%`` (modulo division) can be used but ``/`` (true
division) cannot. Since :ref:`QQ` is a :py:class:`~.Field` its elements
can be used with ``/`` but ``//`` and ``%`` should not be used. Some
domains have a :py:meth:`~.Domain.gcd` method.

>>> ZZ(2) + ZZ(3)
5
>>> ZZ(5) // ZZ(2)
2
>>> ZZ(5) % ZZ(2)
1
>>> QQ(1, 2) / QQ(2, 3)
3/4
>>> ZZ.gcd(ZZ(4), ZZ(2))
2
>>> QQ.gcd(QQ(2,7), QQ(5,3))
1/21
>>> ZZ.is_Field
False
>>> QQ.is_Field
True

There are also many other domains including:

    1. :ref:`GF(p)` for finite fields of prime order.
    2. :ref:`RR` for real (floating point) numbers.
    3. :ref:`CC` for complex (floating point) numbers.
    4. :ref:`QQ(a)` for algebraic number fields.
    5. :ref:`K[x]` for polynomial rings.
    6. :ref:`K(x)` for rational function fields.
    7. :ref:`EX` for arbitrary expressions.

Each domain is represented by a domain object and also an implementation
class (``dtype``) for the elements of the domain. For example the
:ref:`K[x]` domains are represented by a domain object which is an
instance of :py:class:`~.PolynomialRing` and the elements are always
instances of :py:class:`~.PolyElement`. The implementation class
represents particular types of mathematical expressions in a way that is
more efficient than a normal SymPy expression which is of type
:py:class:`~.Expr`. The domain methods :py:meth:`~.Domain.from_sympy` and
:py:meth:`~.Domain.to_sympy` are used to convert from :py:class:`~.Expr`
to a domain element and vice versa.

>>> from sympy import Symbol, ZZ, Expr
>>> x = Symbol('x')
>>> K = ZZ[x]           # polynomial ring domain
>>> K
ZZ[x]
>>> type(K)             # class of the domain
<class 'sympy.polys.domains.polynomialring.PolynomialRing'>
>>> K.dtype             # doctest: +SKIP
<class 'sympy.polys.rings.PolyElement'>
>>> p_expr = x**2 + 1   # Expr
>>> p_expr
x**2 + 1
>>> type(p_expr)
<class 'sympy.core.add.Add'>
>>> isinstance(p_expr, Expr)
True
>>> p_domain = K.from_sympy(p_expr)
>>> p_domain            # domain element
x**2 + 1
>>> type(p_domain)
<class 'sympy.polys.rings.PolyElement'>
>>> K.to_sympy(p_domain) == p_expr
True

The :py:meth:`~.Domain.convert_from` method is used to convert domain
elements from one domain to another.

>>> from sympy import ZZ, QQ
>>> ez = ZZ(2)
>>> eq = QQ.convert_from(ez, ZZ)
>>> type(ez)  # doctest: +SKIP
<class 'int'>
>>> type(eq)  # doctest: +SKIP
<class 'sympy.polys.domains.pythonrational.PythonRational'>

Elements from different domains should not be mixed in arithmetic or other
operations: they should be converted to a common domain first.  The domain
method :py:meth:`~.Domain.unify` is used to find a domain that can
represent all the elements of two given domains.

>>> from sympy import ZZ, QQ, symbols
>>> x, y = symbols('x, y')
>>> ZZ.unify(QQ)
QQ
>>> ZZ[x].unify(QQ)
QQ[x]
>>> ZZ[x].unify(QQ[y])
QQ[x,y]

If a domain is a :py:class:`~.Ring` then is might have an associated
:py:class:`~.Field` and vice versa. The :py:meth:`~.Domain.get_field` and
:py:meth:`~.Domain.get_ring` methods will find or create the associated
domain.

>>> from sympy import ZZ, QQ, Symbol
>>> x = Symbol('x')
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
>>> QQ.has_assoc_Ring
True
>>> QQ.get_ring()
ZZ
>>> K = QQ[x]
>>> K
QQ[x]
>>> K.get_field()
QQ(x)

See also
========

DomainElement: abstract base class for domain elements
construct_domain: construct a minimal domain for some expressions

Nz