/usr/share/axiom-20170501/src/algebra/PADICCT.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | )abbrev category PADICCT PAdicIntegerCategory
++ Author: Clifton J. Williamson
++ Date Created: 15 May 1990
++ Date Last Updated: 15 May 1990
++ Description:
++ This is the category of stream-based representations of
++ the p-adic integers.
PAdicIntegerCategory(p) : Category == SIG where
p : Integer
I ==> Integer
NNI ==> NonNegativeInteger
ST ==> Stream
SUP ==> SparseUnivariatePolynomial
SIG ==> Join(EuclideanDomain,CharacteristicZero) with
digits : % -> ST I
++ \spad{digits(x)} returns a stream of p-adic digits of x.
order : % -> NNI
++ \spad{order(x)} returns the exponent of the highest power of p
++ dividing x.
extend : (%,I) -> %
++ \spad{extend(x,n)} forces the computation of digits up to order n.
complete : % -> %
++ \spad{complete(x)} forces the computation of all digits.
modulus : () -> I
++ \spad{modulus()} returns the value of p.
moduloP : % -> I
++ \spad{modulo(x)} returns a, where \spad{x = a + b p}.
quotientByP : % -> %
++ \spad{quotientByP(x)} returns b, where \spad{x = a + b p}.
approximate : (%,I) -> I
++ \spad{approximate(x,n)} returns an integer y such that
++ \spad{y = x (mod p^n)}
++ when n is positive, and 0 otherwise.
sqrt : (%,I) -> %
++ \spad{sqrt(b,a)} returns a square root of b.
++ Argument \spad{a} is a square root of b \spad{(mod p)}.
root : (SUP I,I) -> %
++ \spad{root(f,a)} returns a root of the polynomial \spad{f}.
++ Argument \spad{a} must be a root of \spad{f} \spad{(mod p)}.
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