Introduction

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 = FlintPadicField

so 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 field 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.

# Nemo.FlintPadicField — Method.

FlintPadicField(p::Integer, 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.

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It is also possible to call the inner constructor directly. It has the following form.

FlintPadicField(p::fmpz, 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.

R = PadicField(7, 30)
S = PadicField(ZZ(65537), 30)

a = R()
b = S(1)
c = S(ZZ(123))
d = R(ZZ(1)//7^2)

P-adic field element constructors

Once a $p$-adic field is constructed, there are various ways to construct elements in that field.

Apart from coercing elements into the $p$-adic field as above, we offer the following functions.

# Base.zero — Method.

zero(R::FlintPadicField)

Return zero in the given $p$-adic field, to the default precision.

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# Base.one — Method.

one(R::FlintPadicField)

Return zero in the given $p$-adic field, to the default precision.

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Elements can also be constructed using the big-oh notation. For this purpose we define the following functions.

# Nemo.O — Method.

O(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).

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# Nemo.O — Method.

O(R::FlintPadicField, m::fmpz)

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).

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# Nemo.O — Method.

O(R::FlintPadicField, m::fmpq)

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).

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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.

Here are some examples of constructing $p$-adic field elements.

R = PadicField(7, 30)
S = PadicField(fmpz(65537), 30)

a = one(R)
b = zero(S)
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 \code{Int} values are used instead of Flint integers or Julia bignums, overflow may result in evaluating the value.

Basic functionality

The following basic functionality is provided by the default $p$-adic field implementation in Nemo, to support construction of generic rings over $p$-adic fields. Any custom $p$-adic field implementation in Nemo should provide these functions along with the usual arithmetic operations.

parent_type(::Type{padic})

Gives the type of the parent object of a Flint $p$-adic field element.

elem_type(R::FlintPadicField)

Given the parent object for a $p$-adic field, return the type of elements of the field.

Base.hash(a::padic, h::UInt)

Return a UInt hexadecimal hash of the $p$-adic field element $a$. This should be xor'd with a fixed random hexadecimal specific to the $p$-adic field type. The hash of the representative of the $p$-adic field element (lifted to $\mathbb{Q}$ and the prime $p$ for the field, should be xor'd with the supplied parameter h as part of computing the hash.

deepcopy(a::padic)

Construct a copy of the given $p$-adic field element and return it. This function must recursively construct copies of all of the internal data in the given element. Nemo $p$-adic field elements are mutable and so returning shallow copies is not sufficient.

mul!(c::padic, a::padic, b::padic)

Multiply $a$ by $b$ and set the existing $p$-adic field element $c$ to the result. This function is provided for performance reasons as it saves allocating a new object for the result and eliminates associated garbage collection.

addeq!(c::padic, a::padic)

In-place addition. Adds $a$ to $c$ and sets $c$ to the result. This function is provided for performance reasons as it saves allocating a new object for the result and eliminates associated garbage collection.

Given the parent object R for a $p$-adic field, the following coercion functions are provided to coerce various elements into the $p$-adic field. Developers provide these by overloading the call operator for the $p$-adic field parent objects.

R()

Coerce zero into the $p$-adic field.

R(n::Integer)
R(f::fmpz)
R(f::fmpq)

Coerce an integer or rational value into the $p$-adic field.

R(f::padic)

Take a $p$-adic field element that is already in the $p$-adic field and simply return it. A copy of the original is not made.

In addition to the above, developers of custom $p$-adic field types must ensure that they provide the equivalent of the function base_ring(R::FlintPadicField) which should return Union{}. In addition to this they should ensure that each $p$-adic field element contains a field parent specifying the parent object of the $p$-adic field element, or at least supply the equivalent of the function parent(a::padic) to return the parent object of a $p$-adic field element.

Basic manipulation

Numerous functions are provided to manipulate $p$-adic field elements. Also see the section on basic functionality above.

# Nemo.base_ring — Method.

base_ring(a::FlintPadicField)

Returns Union{} as this field is not dependent on another field.

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# Nemo.base_ring — Method.

base_ring(a::padic)

Returns Union{} as this field is not dependent on another field.

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# Base.parent — Method.

parent(a::padic)

Returns the parent of the given p-adic field element.

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# Nemo.iszero — Method.

iszero(a::padic)

Return true if the given p-adic field element is zero, otherwise return false.

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# Nemo.isone — Method.

isone(a::padic)

Return true if the given p-adic field element is one, otherwise return false.

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# Nemo.isunit — Method.

isunit(a::padic)

Return true if the given p-adic field element is invertible, i.e. nonzero, otherwise return false.

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# Nemo.prime — Method.

prime(R::FlintPadicField)

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

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# Base.precision — Method.

precision(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 $n$.

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# Nemo.valuation — Method.

valuation(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 $n$.

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# Nemo.lift — Method.

lift(R::FlintIntegerRing, a::padic)

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

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# Nemo.lift — Method.

lift(R::FlintRationalField, a::padic)

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

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Here are some examples of basic manipulation of $p$-adic field elements.

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)

d = one(R)
f = zero(R)
g = isone(d)
h = iszero(f)
k = precision(a)
m = prime(R)
n = valuation(b)
p = lift(FlintZZ, a)
q = lift(FlintQQ, divexact(a, b))

Arithmetic operations

Nemo provides all the standard field operations for $p$-adic field elements, as follows.

Function	Operation
-(a::padic)	unary minus
+(a::padic, b::padic)	addition
-(a::padic, b::padic)	subtraction
*(a::padic, b::padic)	multiplication
divexact(a::padic, b::padic)	exact division

In addition, the following ad hoc field operations are defined.

Function	Operation
+(a::padic, b::Integer)	addition
+(a::Integer, b::padic)	addition
+(a::padic, b::fmpz)	addition
+(a::fmpz, b::padic)	addition
+(a::padic, b::fmpq)	addition
+(a::fmpq, b::padic)	addition
-(a::padic, b::Integer)	subtraction
-(a::Integer, b::padic)	subtraction
-(a::padic, b::fmpz)	subtraction
-(a::fmpz, b::padic)	subtraction
-(a::padic, b::fmpq)	subtraction
-(a::fmpq, b::padic)	subtraction
*(a::padic, b::Integer)	multiplication
*(a::Integer, b::padic)	multiplication
*(a::padic, b::fmpz)	multiplication
*(a::fmpz, b::padic)	multiplication
*(a::padic, b::fmpq)	multiplication
*(a::fmpq, b::padic)	multiplication
divexact(a::padic, b::Integer)	exact division
divexact(a::padic, b::fmpz)	exact division
divexact(a::padic, b::fmpq)	exact division
divexact(a::Integer, b::padic)	exact division
divexact(a::fmpz, b::padic)	exact division
divexact(a::fmpq, b::padic)	exact division
^(a::padic, b::Int)	powering

Here are some examples of arithmetic operations on $p$-adic field elements.

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 = O(R, 7^3)
d = R(2)

f = a + b
g = a - b
h = a*b
j = b*c
k = a*d
m = a + 2
n = 3 - b
p = a*ZZ(5)
q = ZZ(3)*c
r = 2*d
s = 2 + d
t = d - ZZ(2)
u = a + ZZ(1)//7^2
v = (ZZ(12)//11)*b
w = c*(ZZ(1)//7)

Comparison

Nemo provides the comparison operation == for $p$-adic field elements. Julia then automatically provides the corresponding != operation. Here are the functions provided.

Function

==(a::padic, b::padic) isequal(a::padic, b::padic)

Note that == returns true if its arguments are arithmetically equal to the minimum of the two precisions. The isequal function requires them to both be the same precision, as for power series.

In addition, the following ad hoc comparisons are provided, Julia again providing the corresponding != operators.

Function

==(a::padic, b::Integer) ==(a::padic, b::fmpz) ==(a::padic, b::fmpq) ==(a::Integer, b::padic) ==(a::fmpz, b::padic) ==(a::fmpq, b::padic)

Here are some examples of comparisons.

R = PadicField(7, 30)

a = 1 + 2*7 + 4*7^2 + O(R, 7^3)
b = 3*7^3 + O(R, 7^5)
c = O(R, 7^3)
d = R(2)

a == 1 + 2*7 + O(R, 7^2)
b == c
isequal(a, b)
c == R(0)
d == R(2)
ZZ(3) == d
ZZ(3)//7 == c

Inversion

# Base.inv — Method.

inv(a::padic)

Returns $a^{-1}$. If $a = 0$ a DivideError() is thrown.

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Here are some examples of inversion.

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 = 7 + 2*7^2 + O(R, 7^5)

f = inv(a)
g = inv(b)
h = inv(c)
k = inv(d)
l = inv(R(1))

Divisibility

# Nemo.divides — Method.

divides(f::padic, g::padic)

Returns a pair consisting of a flag which is set to true if $g$ divides $f$ and false otherwise, and a value $h$ such that $f = gh$ if such a value exists. If not, the value of $h$ is undetermined.

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Here are some examples of divisibility testing.

R = PadicField(7, 30)

a = 1 + 7 + 2*7^2 + O(R, 7^3)
b = 2 + 3*7 + O(R, 7^5)

flag, q = divides(a, b)

GCD

# Base.gcd — Method.

gcd(x::padic, y::padic)

Returns the greatest common divisor of $x$ and $y$, i.e. the function returns $1$ unless both $a$ and $b$ are $0$, in which case it returns $0$.

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Here are some examples of greatest common divisor.

R = PadicField(7, 30)

a = 1 + 7 + 2*7^2 + O(R, 7^3)
b = 2 + 3*7 + O(R, 7^5)

d = gcd(a, b)
f = gcd(R(0), R(0))

Square root

# Base.sqrt — Method.

sqrt(a::padic)

Return the $p$-adic square root of $a$. We define this only when the valuation of $a$ is even. The precision of the output will be precision$(a) -$ valuation$(a)/2$. If the square root does not exist, an exception is thrown.

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Here are some examples of taking the square root.

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

# Base.exp — Method.

exp(a::padic)

Return the $p$-adic exponential of $a$. We define this only when the valuation of $a$ is positive (unless $a = 0$). The precision of the output will be the same as the precision of the input. If the input is not valid an exception is thrown.

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# Base.log — Method.

log(a::padic)

Return the $p$-adic logarithm of $a$. We define this only when the valuation of $a$ is zero (but not for $a == 0$). The precision of the output will be the same as the precision of the input. If the input is not valid an exception is thrown.

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# Nemo.teichmuller — Method.

teichmuller(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.

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R = PadicField(7, 30)

a = 1 + 7 + 27^2 + O(R, 7^3) b = 2 + 57 + 37^2 + O(R, 7^3) c = 37 + 2*7^2 + O(R, 7^5)

c = exp(c) d = log(a) c = exp(R(0)) d = log(R(1)) f = teichmuller(b)