Qadics
Q-adic fields, that is, unramified extensions of p-adic fields, are provided in Nemo by Flint. This allows construction of $q$-adic fields for any prime power $q$.
Q-adic fields are constructed using the FlintQadicField function. However, for convenience we define
QadicField = FlintQadicFieldso that $q$-adic fields can be constructed using QadicField rather than FlintQadicField. Note that this is the name of the constructor, but not of qadic field type.
The types of $q$-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 | $\mathbb{Q}_q$ | qadic | QadicField |
All the $q$-adic field types belong to the Field abstract type and the $q$-adic field element types belong to the FieldElem abstract type.
P-adic functionality
Q-adic fields in Nemo implement the AbstractAlgebra.jl field interface.
https://nemocas.github.io/AbstractAlgebra.jl/fields.html
Below, we document all the additional function that is provide by Nemo for p-adic fields.
Constructors
In order to construct $q$-adic field elements in Nemo, one must first construct the $q$-adic field itself. This is accomplished with one of the following constructors.
Nemo.FlintQadicField — Method.FlintQadicField(p::Integer, d::Int, prec::Int)Returns the parent object for the $q$-adic field for given prime $p$ and degree $d$, 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.
FlintQadicField(p::fmpz, d::Int, prec::Int)Returns the parent object for the $q$-adic field for given prime $p$ and degree $d$, where the default absolute precision of elements of the field is given by prec.
Here are some examples of creating $q$-adic fields and making use of the resulting parent objects to coerce various elements into those fields.
Examples
R = QadicField(7, 1, 30)
S = QadicField(ZZ(65537), 1, 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.
AbstractAlgebra.Generic.O — Method.O(R::FlintQadicField, 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).
AbstractAlgebra.Generic.O — Method.O(R::FlintQadicField, 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).
AbstractAlgebra.Generic.O — Method.O(R::FlintQadicField, 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).
The $O(p^n)$ construction can be used to construct $q$-adic values of precision $n$ by adding it to integer values representing the $q$-adic value modulo $p^n$ as in the examples.
Examples
R = QadicField(7, 30)
S = QadicField(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
Nemo.prime — Method.prime(R::FlintQadicField)Return the prime $q$ for the given $q$-adic field.
Base.precision — Method.precision(a::qadic)Return the precision of the given $q$-adic field element, i.e. if the element is known to $O(p^n)$ this function will return $n$.
AbstractAlgebra.Generic.valuation — Method.valuation(a::qadic)Return the valuation of the given $q$-adic field element, i.e. if the given element is divisible by $p^n$ but not a higher power of $q$ then the function will return $n$.
Nemo.lift — Method.lift(R::FmpqPolyRing, a::qadic)Return a lift of the given $q$-adic field element to $\mathbb{Q}[x]$.
Nemo.lift — Method.lift(R::FmpzPolyRing, a::qadic)Return a lift of the given $q$-adic field element to $\mathbb{Z}[x]$ if possible.
Examples
R = QadicField(7, 1, 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)
Qx, x = FlintQQ["x"]
p = lift(Qx, a)
Zy, y = FlintZZ["y"]
q = lift(Zy, divexact(a, b))Square root
Base.sqrt — Method.sqrt(a::qadic)Return the $q$-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.
Examples
R = QadicField(7, 1, 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::qadic)Return the $q$-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.
Base.log — Method.log(a::qadic)Return the $q$-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.
Nemo.teichmuller — Method.teichmuller(a::qadic)Return the Teichmuller lift of the $q$-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 $q$ we return zero. If the input is not valid an exception is thrown.
Nemo.frobenius — Method.frobenius(a::qadic, e::Int = 1)Return the image of the $e$-th power of Frobenius on the $q$-adic value $a$. The precision of the output will be the same as the precision of the input.
Examples
R = QadicField(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)
g = frobenius(a, 2)