Padics
P-adic fields are provided in Nemo by Flint. This allows construction of $p$-adic fields for any prime $p$.
P-adic fields are constructed using the FlintPadicField function. However, for convenience we define
PadicField = FlintPadicFieldso that $p$-adic fields can be constructed using PadicField rather than FlintPadicField. Note that this is the name of the constructor, but not of padic field type.
The types of $p$-adic fields in Nemo are given in the following table, along with the libraries that provide them and the associated types of the parent objects.
| Library | Field | Element type | Parent type |
|---|---|---|---|
| Flint | $\mathbb{Q}_p$ | padic | PadicField |
All the $p$-adic field types belong to the Field abstract type and the $p$-adic field element types belong to the FieldElem abstract type.
P-adic functionality
P-adic fields in Nemo implement all the AbstractAlgebra field functionality:.
https://nemocas.github.io/AbstractAlgebra.jl/stable/field
Below, we document all the additional function that is provide by Nemo for p-adic fields.
Constructors
In order to construct $p$-adic field elements in Nemo, one must first construct the $p$-adic field itself. This is accomplished with one of the following constructors.
FlintPadicField — MethodFlintPadicField(p::Integer, prec::Int; kw...)Returns the parent object for the $p$-adic field for given prime $p$, where the default absolute precision of elements of the field is given by prec.
It is also possible to call the inner constructor directly. It has the following form.
FlintPadicField(p::ZZRingElem, prec::Int)Returns the parent object for the $p$-adic field for given prime $p$, where the default absolute precision of elements of the field is given by prec.
Here are some examples of creating $p$-adic fields and making use of the resulting parent objects to coerce various elements into those fields.
Examples
R = PadicField(7, 30)
S = PadicField(ZZ(65537), 30)
a = R()
b = S(1)
c = S(ZZ(123))
d = R(ZZ(1)//7^2)Big-oh notation
Elements of p-adic fields can be constructed using the big-oh notation. For this purpose we define the following functions.
O — MethodO(R::FlintPadicField, m::Integer)Construct the value $0 + O(p^n)$ given $m = p^n$. An exception results if $m$ is not found to be a power of p = prime(R).
O — MethodO(R::FlintPadicField, m::ZZRingElem)Construct the value $0 + O(p^n)$ given $m = p^n$. An exception results if $m$ is not found to be a power of p = prime(R).
O — MethodO(R::FlintPadicField, m::QQFieldElem)Construct the value $0 + O(p^n)$ given $m = p^n$. An exception results if $m$ is not found to be a power of p = prime(R).
The $O(p^n)$ construction can be used to construct $p$-adic values of precision $n$ by adding it to integer values representing the $p$-adic value modulo $p^n$ as in the examples.
Examples
R = PadicField(7, 30)
S = PadicField(ZZ(65537), 30)
c = 1 + 2*7 + 4*7^2 + O(R, 7^3)
d = 13 + 357*ZZ(65537) + O(S, ZZ(65537)^12)
f = ZZ(1)//7^2 + ZZ(2)//7 + 3 + 4*7 + O(R, 7^2)Beware that the expression 1 + 2*p + 3*p^2 + O(R, p^n) is actually computed as a normal Julia expression. Therefore if {Int} values are used instead of Flint integers or Julia bignums, overflow may result in evaluating the value.
Basic manipulation
prime — Methodprime(R::FlintPadicField)Return the prime $p$ for the given $p$-adic field.
precision — Methodprecision(a::padic)Return the precision of the given $p$-adic field element, i.e. if the element is known to $O(p^n)$ this function will return $n$.
valuation — Methodvaluation(a::padic)Return the valuation of the given $p$-adic field element, i.e. if the given element is divisible by $p^n$ but not a higher power of $p$ then the function will return $n$.
lift — Methodlift(R::ZZRing, a::padic)Return a lift of the given $p$-adic field element to $\mathbb{Z}$.
lift — Methodlift(R::QQField, a::padic)Return a lift of the given $p$-adic field element to $\mathbb{Q}$.
Examples
R = PadicField(7, 30)
a = 1 + 2*7 + 4*7^2 + O(R, 7^3)
b = 7^2 + 3*7^3 + O(R, 7^5)
c = R(2)
k = precision(a)
m = prime(R)
n = valuation(b)
p = lift(FlintZZ, a)
q = lift(FlintQQ, divexact(a, b))Square root
sqrt — Methodsqrt(a::Generic.PuiseuxSeriesElem{T}; check::Bool=true) where T <: RingElementReturn the square root of the given Puiseux series $a$. By default the function will throw an exception if the input is not square. If check=false this test is omitted.
Base.sqrt(f::PolyRingElem{T}; check::Bool=true) where T <: RingElementReturn the square root of $f$. By default the function checks the input is square and raises an exception if not. If check=false this check is omitted.
Base.sqrt(a::FracElem{T}; check::Bool=true) where T <: RingElemReturn the square root of $a$. By default the function will throw an exception if the input is not square. If check=false this test is omitted.
sqrt(a::FieldElem)Return the square root of the element a. By default the function will throw an exception if the input is not square. If check=false this test is omitted.
Examples
R = PadicField(7, 30)
a = 1 + 7 + 2*7^2 + O(R, 7^3)
b = 2 + 3*7 + O(R, 7^5)
c = 7^2 + 2*7^3 + O(R, 7^4)
d = sqrt(a)
f = sqrt(b)
f = sqrt(c)
g = sqrt(R(121))Special functions
exp — Methodexp(a::Generic.LaurentSeriesElem)Return the exponential of the power series $a$.
exp(a::Generic.PuiseuxSeriesElem{T}) where T <: RingElementReturn the exponential of the given Puiseux series $a$.
exp(a::AbsPowerSeriesRingElem)Return the exponential of the power series $a$.
exp(a::RelPowerSeriesRingElem)Return the exponential of the power series $a$.
log — Methodlog(a::Generic.PuiseuxSeriesElem{T}) where T <: RingElementReturn the logarithm of the given Puiseux series $a$.
log(a::SeriesElem{T}) where T <: FieldElementReturn the logarithm of the power series $a$.
teichmuller — Methodteichmuller(a::padic)Return the Teichmuller lift of the $p$-adic value $a$. We require the valuation of $a$ to be nonnegative. The precision of the output will be the same as the precision of the input. For convenience, if $a$ is congruent to zero modulo $p$ we return zero. If the input is not valid an exception is thrown.
Examples
R = PadicField(7, 30)
a = 1 + 7 + 2*7^2 + O(R, 7^3)
b = 2 + 5*7 + 3*7^2 + O(R, 7^3)
c = 3*7 + 2*7^2 + O(R, 7^5)
c = exp(c)
d = log(a)
c = exp(R(0))
d = log(R(1))
f = teichmuller(b)