\(p\)-Adic Base Generic Element

\(p\)-Adic Base Generic Element

A common superclass for features shared among all elements of \(\mathbb{Z}_p\) and \(\mathbb{Q}_p\) (regardless of implementation).

AUTHORS:

• David Roe

class sage.rings.padics.padic_base_generic_element.pAdicBaseGenericElement

Bases: sage.rings.padics.padic_generic_element.pAdicGenericElement

Initialization

EXAMPLES:

sage: R = Zp(5); a = R(1287) #indirect doctest

frobenius(arithmetic=True)

Applies a Frobenius automorphism to this element. This is the identity map, since this element lies in \(\QQ_p\); it exists for compatibility with the frobenius() method of elements of extensions of \(\QQ_p\).

INPUT:

• self – a \(p\)-adic element
• arithmetic – whether to apply arithmetic Frobenius (as opposed to geometric Frobenius) – ignored, since both are the identity map anyway.

OUTPUT:

• returns self.

EXAMPLES:

sage: Qp(7)(2).frobenius()
2 + O(7^20)

minimal_polynomial(name)

Returns a minimal polynomial of this \(p\)-adic element, i.e., x - self

INPUT:

• self – a \(p\)-adic element
• name – string: the name of the variable

OUTPUT:

• polynomial – a minimal polynomial of this \(p\)-adic element, i.e., x - self

EXAMPLES:

sage: Zp(5,5)(1/3).minimal_polynomial('x')
(1 + O(5^5))*x + (3 + 5 + 3*5^2 + 5^3 + 3*5^4 + O(5^5))

norm(ground=None)

Returns the norm of this \(p\)-adic element over the ground ring.

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.

INPUT:

• self – a \(p\)-adic element
• ground – a subring of the ground ring (default: base ring)

OUTPUT:

• element – the norm of this \(p\)-adic element over the ground ring

EXAMPLES:

sage: Zp(5)(5).norm()
5 + O(5^21)

trace(ground=None)

Returns the trace of this \(p\)-adic element over the ground ring

INPUT:

• self – a \(p\)-adic element
• ground – a subring of the ground ring (default: base ring)

OUTPUT:

• element – the trace of this \(p\)-adic element over the ground ring

EXAMPLES:

sage: Zp(5,5)(5).trace()
5 + O(5^6)