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.

LibraryFieldElement typeParent type
Flint$\mathbb{Q}_p$PadicFieldElemPadicField

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 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_fieldFunction
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.

source

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)

julia> S = padic_field(ZZ(65537), precision = 30)

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.

OMethod
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).

source
OMethod
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).

source
OMethod
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).

source

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)

julia> S = padic_field(ZZ(65537), precision = 30)

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

primeMethod
prime(R::PadicField)

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

source
precisionMethod
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 $n$.

source
valuationMethod
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 $n$.

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

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

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

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

source

Examples

julia> R = padic_field(7, precision = 30)

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

sqrtMethod
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.

source
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.

source
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.

source
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.

source

Examples

julia> R = padic_field(7, precision = 30)

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

expMethod
exp(a::PadicFieldElem)

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

source
logMethod
log(a::PadicFieldElem)

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

source
teichmullerMethod
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.

source

Examples

julia> R = padic_field(7, precision = 30)

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)