347 lines
9.0 KiB
Python
347 lines
9.0 KiB
Python
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"""Finite extensions of ring domains."""
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from sympy.polys.domains.domain import Domain
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from sympy.polys.domains.domainelement import DomainElement
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from sympy.polys.polyerrors import (CoercionFailed, NotInvertible,
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GeneratorsError)
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from sympy.polys.polytools import Poly
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from sympy.printing.defaults import DefaultPrinting
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class ExtensionElement(DomainElement, DefaultPrinting):
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"""
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Element of a finite extension.
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A class of univariate polynomials modulo the ``modulus``
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of the extension ``ext``. It is represented by the
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unique polynomial ``rep`` of lowest degree. Both
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``rep`` and the representation ``mod`` of ``modulus``
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are of class DMP.
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"""
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__slots__ = ('rep', 'ext')
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def __init__(self, rep, ext):
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self.rep = rep
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self.ext = ext
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def parent(f):
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return f.ext
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def __bool__(f):
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return bool(f.rep)
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def __pos__(f):
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return f
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def __neg__(f):
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return ExtElem(-f.rep, f.ext)
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def _get_rep(f, g):
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if isinstance(g, ExtElem):
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if g.ext == f.ext:
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return g.rep
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else:
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return None
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else:
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try:
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g = f.ext.convert(g)
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return g.rep
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except CoercionFailed:
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return None
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def __add__(f, g):
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rep = f._get_rep(g)
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if rep is not None:
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return ExtElem(f.rep + rep, f.ext)
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else:
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return NotImplemented
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__radd__ = __add__
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def __sub__(f, g):
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rep = f._get_rep(g)
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if rep is not None:
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return ExtElem(f.rep - rep, f.ext)
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else:
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return NotImplemented
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def __rsub__(f, g):
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rep = f._get_rep(g)
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if rep is not None:
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return ExtElem(rep - f.rep, f.ext)
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else:
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return NotImplemented
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def __mul__(f, g):
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rep = f._get_rep(g)
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if rep is not None:
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return ExtElem((f.rep * rep) % f.ext.mod, f.ext)
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else:
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return NotImplemented
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__rmul__ = __mul__
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def _divcheck(f):
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"""Raise if division is not implemented for this divisor"""
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if not f:
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raise NotInvertible('Zero divisor')
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elif f.ext.is_Field:
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return True
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elif f.rep.is_ground and f.ext.domain.is_unit(f.rep.rep[0]):
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return True
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else:
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# Some cases like (2*x + 2)/2 over ZZ will fail here. It is
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# unclear how to implement division in general if the ground
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# domain is not a field so for now it was decided to restrict the
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# implementation to division by invertible constants.
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msg = (f"Can not invert {f} in {f.ext}. "
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"Only division by invertible constants is implemented.")
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raise NotImplementedError(msg)
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def inverse(f):
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"""Multiplicative inverse.
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Raises
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======
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NotInvertible
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If the element is a zero divisor.
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"""
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f._divcheck()
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if f.ext.is_Field:
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invrep = f.rep.invert(f.ext.mod)
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else:
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R = f.ext.ring
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invrep = R.exquo(R.one, f.rep)
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return ExtElem(invrep, f.ext)
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def __truediv__(f, g):
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rep = f._get_rep(g)
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if rep is None:
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return NotImplemented
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g = ExtElem(rep, f.ext)
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try:
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ginv = g.inverse()
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except NotInvertible:
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raise ZeroDivisionError(f"{f} / {g}")
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return f * ginv
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__floordiv__ = __truediv__
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def __rtruediv__(f, g):
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try:
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g = f.ext.convert(g)
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except CoercionFailed:
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return NotImplemented
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return g / f
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__rfloordiv__ = __rtruediv__
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def __mod__(f, g):
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rep = f._get_rep(g)
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if rep is None:
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return NotImplemented
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g = ExtElem(rep, f.ext)
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try:
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g._divcheck()
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except NotInvertible:
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raise ZeroDivisionError(f"{f} % {g}")
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# Division where defined is always exact so there is no remainder
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return f.ext.zero
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def __rmod__(f, g):
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try:
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g = f.ext.convert(g)
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except CoercionFailed:
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return NotImplemented
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return g % f
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def __pow__(f, n):
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if not isinstance(n, int):
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raise TypeError("exponent of type 'int' expected")
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if n < 0:
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try:
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f, n = f.inverse(), -n
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except NotImplementedError:
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raise ValueError("negative powers are not defined")
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b = f.rep
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m = f.ext.mod
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r = f.ext.one.rep
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while n > 0:
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if n % 2:
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r = (r*b) % m
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b = (b*b) % m
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n //= 2
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return ExtElem(r, f.ext)
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def __eq__(f, g):
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if isinstance(g, ExtElem):
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return f.rep == g.rep and f.ext == g.ext
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else:
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return NotImplemented
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def __ne__(f, g):
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return not f == g
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def __hash__(f):
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return hash((f.rep, f.ext))
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def __str__(f):
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from sympy.printing.str import sstr
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return sstr(f.rep)
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__repr__ = __str__
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@property
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def is_ground(f):
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return f.rep.is_ground
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def to_ground(f):
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[c] = f.rep.to_list()
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return c
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ExtElem = ExtensionElement
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class MonogenicFiniteExtension(Domain):
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r"""
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Finite extension generated by an integral element.
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The generator is defined by a monic univariate
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polynomial derived from the argument ``mod``.
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A shorter alias is ``FiniteExtension``.
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Examples
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========
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Quadratic integer ring $\mathbb{Z}[\sqrt2]$:
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>>> from sympy import Symbol, Poly
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>>> from sympy.polys.agca.extensions import FiniteExtension
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>>> x = Symbol('x')
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>>> R = FiniteExtension(Poly(x**2 - 2)); R
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ZZ[x]/(x**2 - 2)
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>>> R.rank
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2
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>>> R(1 + x)*(3 - 2*x)
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x - 1
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Finite field $GF(5^3)$ defined by the primitive
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polynomial $x^3 + x^2 + 2$ (over $\mathbb{Z}_5$).
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>>> F = FiniteExtension(Poly(x**3 + x**2 + 2, modulus=5)); F
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GF(5)[x]/(x**3 + x**2 + 2)
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>>> F.basis
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(1, x, x**2)
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>>> F(x + 3)/(x**2 + 2)
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-2*x**2 + x + 2
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Function field of an elliptic curve:
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>>> t = Symbol('t')
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>>> FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
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ZZ(x)[t]/(t**2 - x**3 - x + 1)
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"""
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is_FiniteExtension = True
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dtype = ExtensionElement
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def __init__(self, mod):
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if not (isinstance(mod, Poly) and mod.is_univariate):
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raise TypeError("modulus must be a univariate Poly")
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# Using auto=True (default) potentially changes the ground domain to a
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# field whereas auto=False raises if division is not exact. We'll let
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# the caller decide whether or not they want to put the ground domain
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# over a field. In most uses mod is already monic.
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mod = mod.monic(auto=False)
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self.rank = mod.degree()
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self.modulus = mod
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self.mod = mod.rep # DMP representation
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self.domain = dom = mod.domain
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self.ring = mod.rep.ring or dom.old_poly_ring(*mod.gens)
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self.zero = self.convert(self.ring.zero)
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self.one = self.convert(self.ring.one)
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gen = self.ring.gens[0]
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self.symbol = self.ring.symbols[0]
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self.generator = self.convert(gen)
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self.basis = tuple(self.convert(gen**i) for i in range(self.rank))
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# XXX: It might be necessary to check mod.is_irreducible here
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self.is_Field = self.domain.is_Field
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def new(self, arg):
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rep = self.ring.convert(arg)
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return ExtElem(rep % self.mod, self)
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def __eq__(self, other):
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if not isinstance(other, FiniteExtension):
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return False
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return self.modulus == other.modulus
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def __hash__(self):
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return hash((self.__class__.__name__, self.modulus))
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def __str__(self):
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return "%s/(%s)" % (self.ring, self.modulus.as_expr())
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__repr__ = __str__
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def convert(self, f, base=None):
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rep = self.ring.convert(f, base)
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return ExtElem(rep % self.mod, self)
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def convert_from(self, f, base):
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rep = self.ring.convert(f, base)
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return ExtElem(rep % self.mod, self)
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def to_sympy(self, f):
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return self.ring.to_sympy(f.rep)
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def from_sympy(self, f):
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return self.convert(f)
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def set_domain(self, K):
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mod = self.modulus.set_domain(K)
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return self.__class__(mod)
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def drop(self, *symbols):
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if self.symbol in symbols:
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raise GeneratorsError('Can not drop generator from FiniteExtension')
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K = self.domain.drop(*symbols)
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return self.set_domain(K)
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def quo(self, f, g):
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return self.exquo(f, g)
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def exquo(self, f, g):
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rep = self.ring.exquo(f.rep, g.rep)
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return ExtElem(rep % self.mod, self)
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def is_negative(self, a):
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return False
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def is_unit(self, a):
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if self.is_Field:
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return bool(a)
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elif a.is_ground:
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return self.domain.is_unit(a.to_ground())
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FiniteExtension = MonogenicFiniteExtension
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