\(p\)-adic ZZ_pX Element

A common superclass implementing features shared by all elements that use NTL’s ZZ_pX as the fundamental data type.

AUTHORS:

• David Roe

class sage.rings.padics.padic_ZZ_pX_element.pAdicZZpXElement

Bases: pAdicExtElement

Initialization

EXAMPLES:

sage: A = Zp(next_prime(50000),10)
sage: S.<x> = A[]
sage: B.<t> = A.ext(x^2+next_prime(50000)) #indirect doctest

norm(base=None)

Return the absolute or relative norm of this element.

Note

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.

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 \(\QQ_p\) or \(\ZZ_p\).

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)

trace(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 \(\QQ_p\) or \(\ZZ_p\).

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)