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Current File : //usr/local/lib/python3.6/dist-packages/sympy/polys/agca/extensions.py
"""Finite extensions of ring domains."""
from sympy.polys.polyerrors import CoercionFailed, NotInvertible
from sympy.polys.polytools import Poly
from sympy.printing.defaults import DefaultPrinting
class ExtensionElement(DefaultPrinting):
"""
Element of a finite extension.
A class of univariate polynomials modulo the ``modulus``
of the extension ``ext``. It is represented by the
unique polynomial ``rep`` of lowest degree. Both
``rep`` and the representation ``mod`` of ``modulus``
are of class DMP.
"""
__slots__ = ('rep', 'ext')
def __init__(self, rep, ext):
self.rep = rep
self.ext = ext
def __neg__(f):
return ExtElem(-f.rep, f.ext)
def _get_rep(f, g):
if isinstance(g, ExtElem):
if g.ext == f.ext:
return g.rep
else:
return None
else:
try:
g = f.ext.convert(g)
return g.rep
except CoercionFailed:
return None
def __add__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem(f.rep + rep, f.ext)
else:
return NotImplemented
__radd__ = __add__
def __sub__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem(f.rep - rep, f.ext)
else:
return NotImplemented
def __rsub__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem(rep - f.rep, f.ext)
else:
return NotImplemented
def __mul__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem((f.rep*rep) % f.ext.mod, f.ext)
else:
return NotImplemented
__rmul__ = __mul__
def inverse(f):
"""Multiplicative inverse.
Raises
======
NotInvertible
If the element is a zero divisor.
"""
if not f.ext.domain.is_Field:
raise NotImplementedError("base field expected")
return ExtElem(f.rep.invert(f.ext.mod), f.ext)
def _invrep(f, g):
rep = f._get_rep(g)
if rep is not None:
return rep.invert(f.ext.mod)
else:
return None
def __truediv__(f, g):
if not f.ext.domain.is_Field:
return NotImplemented
try:
rep = f._invrep(g)
except NotInvertible:
raise ZeroDivisionError
if rep is not None:
return f*ExtElem(rep, f.ext)
else:
return NotImplemented
__floordiv__ = __truediv__
def __rtruediv__(f, g):
try:
return f.ext.convert(g)/f
except CoercionFailed:
return NotImplemented
__rfloordiv__ = __rtruediv__
def __pow__(f, n):
if not isinstance(n, int):
raise TypeError("exponent of type 'int' expected")
if n < 0:
try:
f, n = f.inverse(), -n
except NotImplementedError:
raise ValueError("negative powers are not defined")
b = f.rep
m = f.ext.mod
r = f.ext.one.rep
while n > 0:
if n % 2:
r = (r*b) % m
b = (b*b) % m
n //= 2
return ExtElem(r, f.ext)
def __eq__(f, g):
if isinstance(g, ExtElem):
return f.rep == g.rep and f.ext == g.ext
else:
return NotImplemented
def __ne__(f, g):
return not f == g
def __hash__(f):
return hash((f.rep, f.ext))
def __str__(f):
from sympy.printing.str import sstr
return sstr(f.rep)
__repr__ = __str__
ExtElem = ExtensionElement
class MonogenicFiniteExtension:
r"""
Finite extension generated by an integral element.
The generator is defined by a monic univariate
polynomial derived from the argument ``mod``.
A shorter alias is ``FiniteExtension``.
Examples
========
Quadratic integer ring $\mathbb{Z}[\sqrt2]$:
>>> from sympy import Symbol, Poly
>>> from sympy.polys.agca.extensions import FiniteExtension
>>> x = Symbol('x')
>>> R = FiniteExtension(Poly(x**2 - 2)); R
ZZ[x]/(x**2 - 2)
>>> R.rank
2
>>> R(1 + x)*(3 - 2*x)
x - 1
Finite field $GF(5^3)$ defined by the primitive
polynomial $x^3 + x^2 + 2$ (over $\mathbb{Z}_5$).
>>> F = FiniteExtension(Poly(x**3 + x**2 + 2, modulus=5)); F
GF(5)[x]/(x**3 + x**2 + 2)
>>> F.basis
(1, x, x**2)
>>> F(x + 3)/(x**2 + 2)
-2*x**2 + x + 2
Function field of an elliptic curve:
>>> t = Symbol('t')
>>> FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
ZZ(x)[t]/(t**2 - x**3 - x + 1)
Notes
=====
``FiniteExtension`` is not a subclass of :class:`~.Domain`. Consequently,
a ``FiniteExtension`` can't currently be used as ``domain`` for the
:class:`~.Poly` class.
"""
def __init__(self, mod):
if not (isinstance(mod, Poly) and mod.is_univariate):
raise TypeError("modulus must be a univariate Poly")
mod, rem = mod.div(mod.LC())
if not rem.is_zero:
raise ValueError("modulus could not be made monic")
self.rank = mod.degree()
self.modulus = mod
self.mod = mod.rep # DMP representation
self.domain = dom = mod.domain
self.ring = mod.rep.ring or dom.old_poly_ring(*mod.gens)
self.zero = self.convert(self.ring.zero)
self.one = self.convert(self.ring.one)
gen = self.ring.gens[0]
self.generator = self.convert(gen)
self.basis = tuple(self.convert(gen**i)
for i in range(self.rank))
def convert(self, f):
rep = self.ring.convert(f)
return ExtElem(rep % self.mod, self)
__call__ = convert
def __str__(self):
return "%s/(%s)" % (self.ring, self.modulus.as_expr())
__repr__ = __str__
FiniteExtension = MonogenicFiniteExtension
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