p-Adic Extension Element.

A common superclass for all elements of extension rings and field of Zp and Qp.

AUTHORS:

  • David Roe
class sage.rings.padics.padic_ext_element.pAdicExtElement
static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
_const_term_test()

Returns the constant term of a polynomial representing self.

This function is mainly for troubleshooting, and the meaning of the return value will depend on whether self is capped relative or otherwise.

EXAMPLES:

sage: R = Zp(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 = W(566)
sage: a._const_term_test()
566
_ext_p_list()

Returns a list of integers (in the Eisenstein case) or a list of lists of integers (in the unramified case). self can be reconstructed as a sum of elements of the list times powers of the uniformiser (in the Eisenstein case), or as a sum of powers of the p times polynomials in the generator (in the unramified case).

Note that zeros are truncated from the returned list, so you must use the valuation() function to completely recover self.

INPUTS:

- pos -- bint.  If True, all integers will be in the range [0,p-1],
  otherwise they will be in the range [(1-p)/2, p/2].

OUTPUT:

- L -- A list of integers or list of lists giving the
  series expansion of self.

EXAMPLES:

sage: R = Zp(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: y = W(775, 19); y
w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19)
sage: y._ext_p_list(True)
[1, 0, 4, 0, 2, 1, 2, 4, 1]
sage: y._ext_p_list(False)
[1, 0, -1, 0, 2, 1, 2, 0, 1]
abs()

Returns the p-adic absolute value of self.

This is normalized so that the absolute value of p is 1/p.

INPUT – prec - Integer. The precision of the real field in which the answer is returned. If None, returns a rational for absolutely unramified fields, or a real with 53 bits of precision if ramified.

EXAMPLES: sage: R = Zp(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: w.abs() 0.724779663677696

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