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 padic_field function.

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	$\mathbb{Q}_p$	`PadicFieldElem`	`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.

padic_field — Function

padic_field(p::Integer; precision::Int=64, cached::Bool=true, check::Bool=true)
padic_field(p::ZZRingElem; precision::Int=64, cached::Bool=true, check::Bool=true)

Return the $p$-adic field for the given prime $p$. The default absolute precision of elements of the field may be set with precision.

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

julia> R = padic_field(7, precision = 30)
Field of 7-adic numbers

julia> S = padic_field(ZZ(65537), precision = 30)
Field of 65537-adic numbers

julia> a = R()
O(7^30)

julia> b = S(1)
65537^0 + O(65537^30)

julia> c = S(ZZ(123))
123*65537^0 + O(65537^30)

julia> d = R(ZZ(1)//7^2)
7^-2 + O(7^28)

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 — Method

O(R::PadicField, 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 — Method

O(R::PadicField, 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 — Method

O(R::PadicField, 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

julia> R = padic_field(7, precision = 30)
Field of 7-adic numbers

julia> S = padic_field(ZZ(65537), precision = 30)
Field of 65537-adic numbers

julia> c = 1 + 2*7 + 4*7^2 + O(R, 7^3)
7^0 + 2*7^1 + 4*7^2 + O(7^3)

julia> d = 13 + 357*ZZ(65537) + O(S, ZZ(65537)^12)
13*65537^0 + 357*65537^1 + O(65537^12)

julia> f = ZZ(1)//7^2 + ZZ(2)//7 + 3 + 4*7 + O(R, 7^2)
7^-2 + 2*7^-1 + 3*7^0 + 4*7^1 + O(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 ZZRingElems or Julia BigInts, overflow may result in evaluating the value.

Basic manipulation

prime — Method

prime(R::PadicField)

Return the prime $p$ for the given $p$-adic field.

precision — Method

precision(a::PadicFieldElem)

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 — Method

valuation(a::PadicFieldElem)

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 — Method

lift(R::ZZRing, a::PadicFieldElem)

Return a lift of the given $p$-adic field element to $\mathbb{Z}$.

lift — Method

lift(R::QQField, a::PadicFieldElem)

Return a lift of the given $p$-adic field element to $\mathbb{Q}$.

Examples

julia> R = padic_field(7, precision = 30)
Field of 7-adic numbers

julia> a = 1 + 2*7 + 4*7^2 + O(R, 7^3)
7^0 + 2*7^1 + 4*7^2 + O(7^3)

julia> b = 7^2 + 3*7^3 + O(R, 7^5)
7^2 + 3*7^3 + O(7^5)

julia> c = R(2)
2*7^0 + O(7^30)

julia> k = precision(a)
3

julia> m = prime(R)
7

julia> n = valuation(b)
2

julia> p = lift(ZZ, a)
211

julia> q = lift(QQ, divexact(a, b))
337//49

Square root

sqrt — Method

Base.sqrt(f::PolyRingElem{T}; check::Bool=true) where T <: RingElement

Return 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 <: RingElem

Return 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.

sqrt(a::Generic.PuiseuxSeriesElem{T}; check::Bool=true) where T <: RingElement

Return 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.

Examples

julia> R = padic_field(7, precision = 30)
Field of 7-adic numbers

julia> a = 1 + 7 + 2*7^2 + O(R, 7^3)
7^0 + 7^1 + 2*7^2 + O(7^3)

julia> b = 2 + 3*7 + O(R, 7^5)
2*7^0 + 3*7^1 + O(7^5)

julia> c = 7^2 + 2*7^3 + O(R, 7^4)
7^2 + 2*7^3 + O(7^4)

julia> d = sqrt(a)
7^0 + 4*7^1 + 3*7^2 + O(7^3)

julia> f = sqrt(b)
3*7^0 + 5*7^1 + 7^2 + 7^3 + O(7^5)

julia> f = sqrt(c)
7^1 + 7^2 + O(7^3)

julia> g = sqrt(R(121))
3*7^0 + 5*7^1 + 6*7^2 + 6*7^3 + 6*7^4 + 6*7^5 + 6*7^6 + 6*7^7 + 6*7^8 + 6*7^9 + 6*7^10 + 6*7^11 + 6*7^12 + 6*7^13 + 6*7^14 + 6*7^15 + 6*7^16 + 6*7^17 + 6*7^18 + 6*7^19 + 6*7^20 + 6*7^21 + 6*7^22 + 6*7^23 + 6*7^24 + 6*7^25 + 6*7^26 + 6*7^27 + 6*7^28 + 6*7^29 + O(7^30)

julia> g^2 == R(121)
true

Special functions

exp — Method

exp(a::PadicFieldElem)

Return the $p$-adic exponential of $a$, assuming the $p$-adic exponential function converges at $a$.

log — Method

log(a::PadicFieldElem)

Return the $p$-adic logarithm of $a$, assuming the $p$-adic logarithm converges at $a$.

teichmuller — Method

teichmuller(a::PadicFieldElem)

Return the Teichmuller lift of the $p$-adic value $a$. We require the valuation of $a$ to be non-negative. 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

julia> R = padic_field(7, precision = 30)
Field of 7-adic numbers

julia> a = 1 + 7 + 2*7^2 + O(R, 7^3)
7^0 + 7^1 + 2*7^2 + O(7^3)

julia> b = 2 + 5*7 + 3*7^2 + O(R, 7^3)
2*7^0 + 5*7^1 + 3*7^2 + O(7^3)

julia> c = 3*7 + 2*7^2 + O(R, 7^5)
3*7^1 + 2*7^2 + O(7^5)

julia> c = exp(c)
7^0 + 3*7^1 + 3*7^2 + 4*7^3 + 4*7^4 + O(7^5)

julia> d = log(a)
7^1 + 5*7^2 + O(7^3)

julia> c = exp(R(0))
7^0 + O(7^30)

julia> d = log(R(1))
O(7^30)

julia> f = teichmuller(b)
2*7^0 + 4*7^1 + 6*7^2 + O(7^3)