\(p\)-adic Extension Leaves

The final classes for extensions of \(\ZZ_p\) and \(\QQ_p\) (i.e., classes that are not just designed to be inherited from).

AUTHORS:

• David Roe

class sage.rings.padics.padic_extension_leaves.EisensteinExtensionFieldCappedRelative(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='NTL')

Bases: EisensteinExtensionGeneric, pAdicCappedRelativeFieldGeneric

class sage.rings.padics.padic_extension_leaves.EisensteinExtensionRingCappedAbsolute(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation)

Bases: EisensteinExtensionGeneric, pAdicCappedAbsoluteRingGeneric

class sage.rings.padics.padic_extension_leaves.EisensteinExtensionRingCappedRelative(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='NTL')

Bases: EisensteinExtensionGeneric, pAdicCappedRelativeRingGeneric

class sage.rings.padics.padic_extension_leaves.EisensteinExtensionRingFixedMod(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='NTL')

Bases: EisensteinExtensionGeneric, pAdicFixedModRingGeneric

fraction_field()

Eisenstein extensions with fixed modulus do not support fraction fields.

EXAMPLES:

Sage

sage: S.<x> = ZZ[]
sage: R.<a> = ZpFM(5).extension(x^2 - 5)                                    # needs sage.libs.ntl
sage: R.fraction_field()                                                    # needs sage.libs.ntl
Traceback (most recent call last):
...
TypeError: This implementation of the p-adic ring
does not support fields of fractions.

Python

>>> from sage.all import *
>>> S = ZZ['x']; (x,) = S._first_ngens(1)
>>> R = ZpFM(Integer(5)).extension(x**Integer(2) - Integer(5), names=('a',)); (a,) = R._first_ngens(1)# needs sage.libs.ntl
>>> R.fraction_field()                                                    # needs sage.libs.ntl
Traceback (most recent call last):
...
TypeError: This implementation of the p-adic ring
does not support fields of fractions.

Sage Live

S.<x> = ZZ[]
R.<a> = ZpFM(5).extension(x^2 - 5)                                    # needs sage.libs.ntl
R.fraction_field()                                                    # needs sage.libs.ntl

class sage.rings.padics.padic_extension_leaves.UnramifiedExtensionFieldCappedRelative(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='FLINT')

Bases: UnramifiedExtensionGeneric, pAdicCappedRelativeFieldGeneric

class sage.rings.padics.padic_extension_leaves.UnramifiedExtensionFieldFloatingPoint(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='FLINT')

Bases: UnramifiedExtensionGeneric, pAdicFloatingPointFieldGeneric

class sage.rings.padics.padic_extension_leaves.UnramifiedExtensionRingCappedAbsolute(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='FLINT')

Bases: UnramifiedExtensionGeneric, pAdicCappedAbsoluteRingGeneric

class sage.rings.padics.padic_extension_leaves.UnramifiedExtensionRingCappedRelative(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='FLINT')

Bases: UnramifiedExtensionGeneric, pAdicCappedRelativeRingGeneric

class sage.rings.padics.padic_extension_leaves.UnramifiedExtensionRingFixedMod(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='FLINT')

Bases: UnramifiedExtensionGeneric, pAdicFixedModRingGeneric

class sage.rings.padics.padic_extension_leaves.UnramifiedExtensionRingFloatingPoint(exact_modulus, poly, prec, print_mode, shift_seed, names, implementation='FLINT')

Bases: UnramifiedExtensionGeneric, pAdicFloatingPointRingGeneric

Make live