def _minpoly_pow(ex, pw, x, dom, mp=None):
"""
Returns ``minpoly(ex**pw, x)``
Parameters
==========
ex : algebraic element
pw : rational number
x : indeterminate of the polynomial
dom: ground domain
mp : minimal polynomial of ``p``
Examples
========
>>> from sympy import sqrt, QQ, Rational
>>> from sympy.polys.numberfields import _minpoly_pow, minpoly
>>> from sympy.abc import x, y
>>> p = sqrt(1 + sqrt(2))
>>> _minpoly_pow(p, 2, x, QQ)
x**2 - 2*x - 1
>>> minpoly(p**2, x)
x**2 - 2*x - 1
>>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y))
x**3 - y
>>> minpoly(y**Rational(1, 3), x)
x**3 - y
"""
pw = sympify(pw)
if not mp:
mp = _minpoly_compose(ex, x, dom)
if not pw.is_rational:
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
if pw < 0:
if mp == x:
raise ZeroDivisionError('%s is zero' % ex)
mp = _invertx(mp, x)
if pw == -1:
return mp
pw = -pw
ex = 1/ex
y = Dummy(str(x))
mp = mp.subs({x: y})
n, d = pw.as_numer_denom()
res = Poly(resultant(mp, x**d - y**n, gens=[y]), x, domain=dom)
_, factors = res.factor_list()
res = _choose_factor(factors, x, ex**pw, dom)
return res.as_expr()
def _minpoly_pow(ex, pw, x, dom, mp=None):
"""
Returns ``minpoly(ex**pw, x)``
Parameters
==========
ex : algebraic element
pw : rational number
x : indeterminate of the polynomial
dom: ground domain
mp : minimal polynomial of ``p``
Examples
========
>>> from sympy import sqrt, QQ, Rational
>>> from sympy.polys.numberfields import _minpoly_pow, minpoly
>>> from sympy.abc import x, y
>>> p = sqrt(1 + sqrt(2))
>>> _minpoly_pow(p, 2, x, QQ)
x**2 - 2*x - 1
>>> minpoly(p**2, x)
x**2 - 2*x - 1
>>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y))
x**3 - y
>>> minpoly(y**Rational(1, 3), x)
x**3 - y
"""
pw = sympify(pw)
if not mp:
mp = _minpoly_compose(ex, x, dom)
if not pw.is_rational:
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
if pw < 0:
if mp == x:
raise ZeroDivisionError('%s is zero' % ex)
mp = _invertx(mp, x)
if pw == -1:
return mp
pw = -pw
ex = 1 / ex
y = Dummy(str(x))
mp = mp.subs({x: y})
n, d = pw.as_numer_denom()
res = Poly(resultant(mp, x**d - y**n, gens=[y]), x, domain=dom)
_, factors = res.factor_list()
res = _choose_factor(factors, x, ex**pw, dom)
return res.as_expr()
def _minpoly_pow(ex, pw, x, mp=None):
"""
Returns ``minpoly(ex**pw, x)``
Parameters
==========
p : algebraic number
mp : minimal polynomial of ``p``
pw : rational number
x : indeterminate of the polynomial
Examples
========
>>> from sympy import sqrt
>>> from sympy.polys.numberfields import _minpoly_pow, minpoly
>>> from sympy.abc import x
>>> p = sqrt(1 + sqrt(2))
>>> _minpoly_pow(p, 2, x)
x**2 - 2*x - 1
>>> minpoly(p**2, x)
x**2 - 2*x - 1
"""
pw = sympify(pw)
if not mp:
mp = _minpoly1(ex, x)
if not pw.is_rational:
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
if pw < 0:
if mp == x:
raise ZeroDivisionError('%s is zero' % ex)
mp = _invertx(mp, x)
if pw == -1:
return mp
pw = -pw
ex = 1 / ex
y = Dummy(str(x))
mp = mp.subs({x: y})
n, d = pw.as_numer_denom()
res = resultant(mp, x**d - y**n, gens=[y])
_, factors = factor_list(res)
res = _choose_factor(factors, x, ex**pw)
return res
def _minpoly_pow(ex, pw, x, mp=None):
"""
Returns ``minpoly(ex**pw, x)``
Parameters
==========
p : algebraic number
mp : minimal polynomial of ``p``
pw : rational number
x : indeterminate of the polynomial
Examples
========
>>> from sympy import sqrt
>>> from sympy.polys.numberfields import _minpoly_pow, minpoly
>>> from sympy.abc import x
>>> p = sqrt(1 + sqrt(2))
>>> _minpoly_pow(p, 2, x)
x**2 - 2*x - 1
>>> minpoly(p**2, x)
x**2 - 2*x - 1
"""
pw = sympify(pw)
if not mp:
mp = _minpoly1(ex, x)
if not pw.is_rational:
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
if pw < 0:
if mp == x:
raise ZeroDivisionError('%s is zero' % ex)
mp = _invertx(mp, x)
if pw == -1:
return mp
pw = -pw
ex = 1/ex
y = Dummy(str(x))
mp = mp.subs({x:y})
n, d = pw.as_numer_denom()
res = resultant(mp, x**d - y**n, gens=[y])
_, factors = factor_list(res)
res = _choose_factor(factors, x, ex**pw)
return res