o Rh @sDdZddlmZddlmZmZddlmZeGdddeZdS)z(Implementation of :class:`Field` class. )Ring) NotReversible DomainError)publicc@speZdZdZdZdZddZddZddZd d Z d d Z d dZ ddZ ddZ ddZddZddZdS)FieldzRepresents a field domain. TcCs td|)z)Returns a ring associated with ``self``. z#there is no ring associated with %s)rselfr [/home/air/sanwanet/backup_V2/venv/lib/python3.10/site-packages/sympy/polys/domains/field.pyget_rings zField.get_ringcCs|S)z*Returns a field associated with ``self``. r rr r r get_fieldszField.get_fieldcC||S)z=Exact quotient of ``a`` and ``b``, implies ``__truediv__``. r rabr r r exquoz Field.exquocCr )z6Quotient of ``a`` and ``b``, implies ``__truediv__``. r rr r r quorz Field.quocCs|jS)z0Remainder of ``a`` and ``b``, implies nothing. zerorr r r remsz Field.remcCs|||jfS)z6Division of ``a`` and ``b``, implies ``__truediv__``. rrr r r div#sz Field.divcCsfz|}Wn ty|jYSw|||||}|||||}||||S)a Returns GCD of ``a`` and ``b``. This definition of GCD over fields allows to clear denominators in `primitive()`. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy import S, gcd, primitive >>> from sympy.abc import x >>> QQ.gcd(QQ(2, 3), QQ(4, 9)) 2/9 >>> gcd(S(2)/3, S(4)/9) 2/9 >>> primitive(2*x/3 + S(4)/9) (2/9, 3*x + 2) )r ronegcdnumerlcmdenomconvertrrrringpqr r r r's   z Field.gcdcCsP|||}||jkr ||jkr|j|j|jfS|j|||fS|||j|fS)zK Returns x, y, g such that a * x + b * y == g == gcd(a, b) )rrr)rrrdr r r gcdexGs   z Field.gcdexcCshz|}Wn ty||YSw|||||}|||||}||||S)z Returns LCM of ``a`` and ``b``. >>> from sympy.polys.domains import QQ >>> from sympy import S, lcm >>> QQ.lcm(QQ(2, 3), QQ(4, 9)) 4/3 >>> lcm(S(2)/3, S(4)/9) 4/3 )r rrrrrrrr r r rUs   z Field.lcmcCs|rd|Std)z!Returns ``a**(-1)`` if possible. zzero is not reversible)rrrr r r revertmsz Field.revertcCst|S)z$Return true if ``a`` is a invertible)boolr%r r r is_unittrz Field.is_unitN)__name__ __module__ __qualname____doc__is_Fieldis_PIDr r rrrrrr#rr&r(r r r r rs  rN) r,sympy.polys.domains.ringrsympy.polys.polyerrorsrrsympy.utilitiesrrr r r r s