A superclass for implementations of and .
AUTHORS:
Bases: sage.rings.padics.padic_generic.pAdicGeneric
Returns the absolute discriminant of this -adic ring
EXAMPLES:
sage: Zp(5).absolute_discriminant()
1
Returns the discriminant of this -adic ring over K
INPUT:
OUTPUT:
EXAMPLES:
sage: Zp(5).discriminant()
1
Returns the fraction field of self.
INPUT:
OUTPUT:
EXAMPLES:
sage: R = Zp(5, print_mode='digits')
sage: K = R.fraction_field(); repr(K(1/3))[3:]
'31313131313131313132'
sage: L = R.fraction_field({'max_ram_terms':4}); repr(L(1/3))[3:]
'3132'
Returns the nth generator of this extension. For base rings/fields, we consider the generator to be the prime.
EXAMPLES:
sage: R = Zp(5); R.gen()
5 + O(5^21)
Returns whether or not has a primitive root of unity.
EXAMPLES:
sage: Zp(2).has_pth_root()
True
sage: Zp(17).has_pth_root()
False
Returns whether or not has a primitive root of unity.
INPUT:
OUTPUT:
EXAMPLES:
sage: R=Zp(37)
sage: R.has_root_of_unity(12)
True
sage: R.has_root_of_unity(11)
False
Returns the integer ring of self, possibly with print_mode changed.
INPUT:
OUTPUT:
EXAMPLES:
sage: K = Qp(5, print_mode='digits')
sage: R = K.integer_ring(); repr(R(1/3))[3:]
'31313131313131313132'
sage: S = K.integer_ring({'max_ram_terms':4}); repr(S(1/3))[3:]
'3132'
Returns whether the Galois group is abelian, i.e. True. #should this be automorphism group?
EXAMPLES:
sage: R = Zp(3, 10,'fixed-mod'); R.is_abelian()
True
Returns whether self and ring are isomorphic, i.e. whether ring is an implementation of for the same prime as self.
INPUT:
OUTPUT:
EXAMPLES:
sage: R = Zp(5, 15, print_mode='digits'); S = Zp(5, 44, print_max_terms=4); R.is_isomorphic(S)
True
Returns whether or not this is a normal extension, i.e. True.
EXAMPLES:
sage: R = Zp(3, 10,'fixed-mod'); R.is_normal()
True
Creates a visualization of this -adic ring as a fractal similar as a generalization of the the Sierpi’nski triangle. The resulting image attempts to capture the algebraic and topological characteristics of .
INPUT:
REFERENCES:
EXAMPLES:
sage: Zp(3).plot()
sage: Zp(5).plot(max_points=625)
sage: Zp(23).plot(rgbcolor=(1,0,0))
Returns a uniformizer for this ring.
EXAMPLES:
sage: R = Zp(3,5,'fixed-mod', 'series')
sage: R.uniformizer()
3 + O(3^5)
Returns the nth power of the uniformizer of self (as an element of self).
EXAMPLES:
sage: R = Zp(5)
sage: R.uniformizer_pow(5)
5^5 + O(5^25)
sage: R.uniformizer_pow(infinity)
0
Returns a generator of the group of roots of unity.
INPUT:
OUTPUT:
EXAMPLES:
sage: R = Zp(37,5)
sage: R.zeta(12)
8 + 24*37 + 37^2 + 29*37^3 + 23*37^4 + O(37^5)
Returns the order of the group of roots of unity.
EXAMPLES:
sage: R = Zp(37); R.zeta_order()
36
sage: Zp(2).zeta_order()
2