# $$p$$-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

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)