p-Adic Extension Element¶
A common superclass for all elements of extension rings and field of \(\ZZ_p\) and \(\QQ_p\).
AUTHORS:
- David Roe (2007): initial version
- Julian Rueth (2012-10-18): added residue
-
class
sage.rings.padics.padic_ext_element.
pAdicExtElement
¶ Bases:
sage.rings.padics.padic_generic_element.pAdicGenericElement
-
frobenius
(arithmetic=True)¶ Returns the image of this element under the Frobenius automorphism applied to its parent.
INPUT:
self
– an element of an unramified extension.arithmetic
– whether to apply the arithmetic Frobenius (acting by raising to the \(p\)-th power on the residue field). IfFalse
is provided, the image of geometric Frobenius (raising to the \((1/p)\)-th power on the residue field) will be returned instead.
EXAMPLES:
sage: R.<a> = Zq(5^4,3) sage: a.frobenius() (a^3 + a^2 + 3*a) + (3*a + 1)*5 + (2*a^3 + 2*a^2 + 2*a)*5^2 + O(5^3) sage: f = R.defining_polynomial() sage: f(a) O(5^3) sage: f(a.frobenius()) O(5^3) sage: for i in range(4): a = a.frobenius() sage: a a + O(5^3) sage: K.<a> = Qq(7^3,4) sage: b = (a+1)/7 sage: c = b.frobenius(); c (3*a^2 + 5*a + 1)*7^-1 + (6*a^2 + 6*a + 6) + (4*a^2 + 3*a + 4)*7 + (6*a^2 + a + 6)*7^2 + O(7^3) sage: c.frobenius().frobenius() (a + 1)*7^-1 + O(7^3)
An error will be raised if the parent of self is a ramified extension:
sage: K.<a> = Qp(5).extension(x^2 - 5) sage: a.frobenius() Traceback (most recent call last): ... NotImplementedError: Frobenius automorphism only implemented for unramified extensions
-
residue
(absprec=1)¶ Reduces this element modulo \(\pi^\mathrm{absprec}\).
INPUT:
absprec
- a non-negative integer (default:1
)
OUTPUT:
This element reduced modulo \(\pi^\mathrm{absprec}\).
If
absprec
is zero, then as an element of \(\ZZ/(1)\).If
absprec
is one, then as an element of the residue field.Note
Only implemented for
absprec
less than or equal to one.AUTHORS:
- Julian Rueth (2012-10-18): initial version
EXAMPLES:
Unramified case:
sage: R = ZpCA(3,5) sage: S.<a> = R[] sage: W.<a> = R.extension(a^2 + 9*a + 1) sage: (a + 1).residue(1) a0 + 1 sage: a.residue(2) Traceback (most recent call last): ... NotImplementedError: reduction modulo p^n with n>1.
Eisenstein case:
sage: R = ZpCA(3,5) sage: S.<a> = R[] sage: W.<a> = R.extension(a^2 + 9*a + 3) sage: (a + 1).residue(1) 1 sage: a.residue(2) Traceback (most recent call last): ... NotImplementedError: residue() not implemented in extensions for absprec larger than one.
TESTS:
sage: K = Qp(3,5) sage: S.<a> = R[] sage: W.<a> = R.extension(a^2 + 9*a + 1) sage: (a/3).residue(0) Traceback (most recent call last): ... ValueError: element must have non-negative valuation in order to compute residue.
sage: R = ZpFM(3,5) sage: S.<a> = R[] sage: W.<a> = R.extension(a^2 + 9*a + 1) sage: W.one().residue(0) 0 sage: a.residue(-1) Traceback (most recent call last): ... ValueError: cannot reduce modulo a negative power of the uniformizer. sage: a.residue(16) Traceback (most recent call last): ... PrecisionError: insufficient precision to reduce modulo p^16.
-