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