Coverage for local/lib/python2.7/site-packages/sage/libs/linkages/padics/unram_shared.pxi : 63%

cimport cython 

@cython.binding(True) 

def frobenius_unram(self, arithmetic=True): 

""" 

Returns the image of this element under the Frobenius automorphism 

applied to its parent. 

INPUT: 

- ``self`` -- an element of an unramified extension. 

- ``arithmetic`` -- whether to apply the arithmetic Frobenius (acting 

by raising to the `p`-th power on the residue field). If ``False`` is 

provided, the image of geometric Frobenius (raising to the `(1/p)`-th 

power on the residue field) will be returned instead. 

EXAMPLES:: 

sage: R.<a> = Zq(5^4,3) 

sage: a.frobenius() 

(a^3 + a^2 + 3*a) + (3*a + 1)*5 + (2*a^3 + 2*a^2 + 2*a)*5^2 + O(5^3) 

sage: f = R.defining_polynomial() 

sage: f(a) 

O(5^3) 

sage: f(a.frobenius()) 

O(5^3) 

sage: for i in range(4): a = a.frobenius() 

sage: a 

a + O(5^3) 

sage: K.<a> = Qq(7^3,4) 

sage: b = (a+1)/7 

sage: c = b.frobenius(); c 

(3*a^2 + 5*a + 1)*7^-1 + (6*a^2 + 6*a + 6) + (4*a^2 + 3*a + 4)*7 + (6*a^2 + a + 6)*7^2 + O(7^3) 

sage: c.frobenius().frobenius() 

(a + 1)*7^-1 + O(7^3) 

An error will be raised if the parent of self is a ramified extension:: 

sage: K.<a> = Qp(5).extension(x^2 - 5) 

sage: a.frobenius() 

Traceback (most recent call last): 

... 

NotImplementedError: Frobenius automorphism only implemented for unramified extensions 

""" 

if self == 0: 

return self 

R = self.parent() 

p = R.prime() 

a = R.gen() 

frob_a = R._frob_gen() 

ppow = self.valuation() 

unit = self.unit_part() 

coefs = unit.expansion() 

ans = 0 

# Xavier's implementation based on Horner scheme 

for i in range(R.f()-1, -1, -1): 

update = 0 

for j in range(len(coefs)-1, -1, -1): 

update *= p 

try: 

update += coefs[j][i] 

except IndexError: 

pass 

ans *= frob_a 

ans += update 

return ans << ppow 

@cython.binding(True) 

def norm_unram(self, base = None): 

""" 

Return the absolute or relative norm of this element. 

.. WARNING:: 

This is not the `p`-adic absolute value. This is a 

field theoretic norm down to a ground ring. If you want the 

`p`-adic absolute value, use the ``abs()`` function instead. 

INPUT: 

``base`` -- a subfield of the parent `L` of this element. 

The norm is the relative norm from ``L`` to ``base``. 

Defaults to the absolute norm down to `\mathbb{Q}_p` or `\mathbb{Z}_p`. 

EXAMPLES:: 

sage: R = ZpCR(5,5) 

sage: S.<x> = R[] 

sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5 

sage: W.<w> = R.ext(f) 

sage: ((1+2*w)^5).norm() 

1 + 5^2 + O(5^5) 

sage: ((1+2*w)).norm()^5 

1 + 5^2 + O(5^5) 

TESTS:: 

sage: R = ZpCA(5,5) 

sage: S.<x> = ZZ[] 

sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5 

sage: W.<w> = R.ext(f) 

sage: ((1+2*w)^5).norm() 

1 + 5^2 + O(5^5) 

sage: ((1+2*w)).norm()^5 

1 + 5^2 + O(5^5) 

sage: R = ZpFM(5,5) 

sage: S.<x> = ZZ[] 

sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5 

sage: W.<w> = R.ext(f) 

sage: ((1+2*w)^5).norm() 

1 + 5^2 + O(5^5) 

sage: ((1+2*w)).norm()^5 

1 + 5^2 + O(5^5) 

TESTS: 

Check that :trac:`11586` has been resolved:: 

sage: R.<x> = QQ[] 

sage: f = x^2 + 3*x + 1 

sage: M.<a> = Qp(7).extension(f) 

sage: M(7).norm() 

7^2 + O(7^22) 

sage: b = 7*a + 35 

sage: b.norm() 

4*7^2 + 7^3 + O(7^22) 

sage: b*b.frobenius() 

4*7^2 + 7^3 + O(7^22) 

""" 

if base is not None: 

if base is self.parent(): 

return self 

else: 

raise NotImplementedError 

if self._is_exact_zero(): 

return self.parent().ground_ring()(0) 

elif self._is_inexact_zero(): 

return self.ground_ring(0, self.valuation()) 

if self.valuation() == 0: 

return self.parent().ground_ring()(self.matrix_mod_pn().det()) 

else: 

if self.parent().e() == 1: 

norm_of_uniformizer = self.parent().ground_ring().uniformizer_pow(self.parent().degree()) 

else: 

norm_of_uniformizer = (-1)**self.parent().degree() * self.parent().defining_polynomial()[0] 

return self.parent().ground_ring()(self.unit_part().matrix_mod_pn().det()) * norm_of_uniformizer**self.valuation() 

@cython.binding(True) 

def trace_unram(self, base = None): 

""" 

Return the absolute or relative trace of this element. 

If ``base`` is given then ``base`` must be a subfield of the 

parent `L` of ``self``, in which case the norm is the relative 

norm from `L` to ``base``. 

In all other cases, the norm is the absolute norm down to 

`\mathbb{Q}_p` or `\mathbb{Z}_p`. 

EXAMPLES:: 

sage: R = ZpCR(5,5) 

sage: S.<x> = R[] 

sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5 

sage: W.<w> = R.ext(f) 

sage: a = (2+3*w)^7 

sage: b = (6+w^3)^5 

sage: a.trace() 

3*5 + 2*5^2 + 3*5^3 + 2*5^4 + O(5^5) 

sage: a.trace() + b.trace() 

4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5) 

sage: (a+b).trace() 

4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5) 

TESTS:: 

sage: R = ZpCA(5,5) 

sage: S.<x> = ZZ[] 

sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5 

sage: W.<w> = R.ext(f) 

sage: a = (2+3*w)^7 

sage: b = (6+w^3)^5 

sage: a.trace() 

3*5 + 2*5^2 + 3*5^3 + 2*5^4 + O(5^5) 

sage: a.trace() + b.trace() 

4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5) 

sage: (a+b).trace() 

4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5) 

sage: R = ZpFM(5,5) 

sage: S.<x> = R[] 

sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5 

sage: W.<w> = R.ext(f) 

sage: a = (2+3*w)^7 

sage: b = (6+w^3)^5 

sage: a.trace() 

3*5 + 2*5^2 + 3*5^3 + 2*5^4 + O(5^5) 

sage: a.trace() + b.trace() 

4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5) 

sage: (a+b).trace() 

4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5) 

""" 

if base is not None: 

if base is self.parent(): 

return self 

else: 

raise NotImplementedError 

if self._is_exact_zero(): 

return self.parent().ground_ring()(0) 

elif self._is_inexact_zero(): 

return self.ground_ring(0, self.precision_absolute()) 

if self.valuation() >= 0: 

return self.parent().ground_ring()(self.matrix_mod_pn().trace()) 

else: 

shift = -self.valuation() 

return self.parent().ground_ring()((self * self.parent().prime() ** shift).matrix_mod_pn().trace()) / self.parent().prime()**shift