Profinite Numbers

Profinite Numbers of Number Fields

Let \(K\) be a number field with ring of integers \(O\). Let \(\hat{O}\) be the ring of profinite \(K\)-integers (cf. profinite_integer). The ring of profinite \(K\)-numbers \(\hat{K}\) is the ring of fractions of \(\hat{O}\) with respect to \(\ZZ \setminus \{0\}\). So a profinite \(K\)-integer is a fraction \(a/b\) with \(a \in \hat{O}\) and \(b \in \ZZ \setminus \{0\}\). With \(\{n\hat{\ZZ} \mid n \in \ZZ_{>0}\}\) as a basis of open neighborhoods of zero, \(\hat{K}\) is a topological ring. It is also a commutative \(K\)-algebra.

In SageMath profinite numbers are implemented in the class ProfiniteNumber as a pair \(a = (n, d)\) where \(n\) is a ProfiniteInteger, called the numerator, and \(d\) is a positive integer, called the denominator. We define the represented subset of \(a\) to be the represented subset of \(n\) divided by \(d\), so an \(\alpha \in \hat{K}\) is represented by \(a\) if and only if \(d \cdot \alpha\) is represented by \(n\).

We define the value of \(a\) to be the value of \(n\) divided by \(d\), which is a \(K\)-element. Likewise we define the modulus of \(a\) to be the modulus of \(n\) divided by \(d\), which is a fractional \(O\)-ideal in \(K\). One can check that the represented subset of \(a\) equals \(x + m \hat{O}\) if \(x\) and \(m\) denote the value and modulus of \(a\) respectively.

We can approximate any \(\alpha \in \hat{K}\) arbitrarily closely by a ProfiniteNumber in the following sense: for any open neighborhood \(U\) of \(\alpha\) in \(\hat{K}\), there exists a ProfiniteNumber representing \(\alpha\) whose represented subset is contained in \(U\).

In the rest of this file we will refer to instances of ProfiniteNumber as profinite numbers. We will refer to exact mathematical profinite \(K\)-numbers as elements of \(\hat{K}\).

Profinite Numbers over \(\QQ\)

In SageMath the ring \(\hat{\QQ}\) is implemented as Qhat:

sage: Qhat
Ring of Profinite Numbers of Rational Field
sage: Qhat is ProfiniteNumbers(QQ)
True

We can create profinite \(\QQ\)-integers by giving a numerator, a profinite \(\QQ\)-integer, and a denominator, a non-zero integer:

sage: a = Qhat(Zhat(3, 10), 7)
sage: a.numerator()
3 mod 10
sage: a.denominator()
7

Such a profinite integer is printed using its value and modulus:

sage: a.value()
3/7
sage: a.modulus()
10/7
sage: a
3/7 mod 10/7

Let’s see which elements of \(\QQ\) of height at most \(30\) are represented by a:

sage: X = sorted(set([m/n for m in srange(-30, 31) for n in srange(1, 31)]))
sage: [x for x in X if a.represents(x)]
[-21, -11, -27/7, -17/7, -1, 3/7, 13/7, 23/7, 9, 19, 29]

The value and modulus of a profinite number uniquely determine its numerator and denominator, because we always keep the denominator as small as possible:

sage: b = Qhat(Zhat(7, 35), 14)
sage: b.numerator() # this is not 7 mod 35
1 mod 5
sage: b.denominator() # this is not 14
2

We can do the reduction above, since (7 mod 35)/14 and (1 mod 5)/2 have the same represented subset.

We can also construct profinite numbers by specifying their value and modulus:

sage: c = Qhat(2/3, 30); c
2/3 mod 30
sage: c.numerator(), c.denominator()
(2 mod 90, 3)

We can perform arithmetic on profinite numbers:

sage: a + b
3/14 mod 5/14
sage: c - b
1/6 mod 5/2
sage: a * c
2/7 mod 10/21
sage: a / 7
3/49 mod 10/49

See _add_(), _sub_(), _mul_(), _div_() for details.

Recall the natural isomorphism \(\hat{\ZZ} \to \prod_p \ZZ_p\) (with \(p\) ranging over all prime numbers and \(\ZZ_p\) denoting the ring of \(p\)-adic integers) induced by the Chinese Remainder Theorem. It can be naturally extended to an isomorphism \(\hat{\QQ} \to \prod_p' \QQ_p\) with \(\QQ_p\) denoting the field of \(p\)-adic numbers and the restricted product taken with respect to the open subrings \(\ZZ_p\). The ring on the right hand side is also known as the finite adèle ring of \(\QQ\). We can take this point of view in SageMath as well:

sage: d_2 = Qp(2)(3/2, 3); d_2
2^-1 + 1 + O(2^3)
sage: d_5 = Qp(5)(10, 2); d_5
2*5 + O(5^2)
sage: d_7 = Qp(7)(2/49, -1); d_7
2*7^-2 + O(7^-1)
sage: d = Qhat([d_2, d_5, d_7]); d
2755/98 mod 200/7
sage: for p in prime_range(10):
....:     print(d[p])
....:
2^-1 + 1 + O(2^3)
O(3^0)
2*5 + O(5^2)
2*7^-2 + O(7^-1)

Note

As \(p\)-adic numbers are currently only fully implemented for rational \(p\), this functionality is only available for profinite \(\QQ\)-numbers, not for profinite \(K\)-numbers for non-trivial number fields \(K\).

Profinite Integers over \(K\)

We also have profinite integers over non-trivial number fields \(K\). Let’s have a look at an example:

sage: K.<a> = NumberField(x^3-2)
sage: Khat = ProfiniteNumbers(K); Khat
Ring of Profinite Numbers of Number Field in a with defining polynomial x^3 - 2
sage: b = Khat.an_element(); b
8/5*a^2 mod (7/5*a^2 - 1/5)

Over general number fields moduli are fractional ideals:

sage: b.value()
8/5*a^2
sage: b.modulus()
Fractional ideal (7/5*a^2 - 1/5)

We can create profinite numbers as before using numerator/denominator pairs or value/modulus pairs:

sage: Ohat = ProfiniteIntegers(K)
sage: c = Khat(Ohat(a, 10), 2); c # numerator/denominator
1/2*a mod (5)
sage: d = Khat(a, 10/3); d # value/modulus
a mod (10/3)

And we can then perform arithmetic with our profinite numbers:

sage: c + d
-1/6*a mod (5/3)
sage: c - d
-1/2*a mod (5/3)
sage: c * d
1/2*a^2 mod (5/3*a)
sage: c / a
1/2 mod (5/2*a^2)

Above, the \(K\)-element a is coerced into Khat as a mod 0, which is a profinite \(K\)-number representing only a. The division is then performed inside Khat.

Profinite \(K\)-integers coerce to profinite \(K\)-numbers with denominator \(1\):

sage: Khat(Ohat(0, a))
0 mod (a)

See also

profinite_integer, adele

REFERENCES:

[Her2021] Mathé Hertogh, Computing with adèles and idèles, master’s thesis, Leiden University, 2021.

This implementation of profinite numbers is based on [Her2021]. An extensive exposition of properties, design choices and two applications can be found there.

AUTHORS:

• Mathé Hertogh (2021-07): initial version based on [Her2021]

class adeles.profinite_number.ProfiniteNumber(parent, numerator, denominator=1)

Bases: sage.structure.element.CommutativeAlgebraElement

Profinite Number of a Number Field

REFERENCES:

Section 3.2 of [Her2021].

__init__(parent, numerator, denominator=1)

Construct the profinite number numerator / denominator

INPUT:

• parent – A ring of profinite numbers over some number field \(K\)

• numerator – a profinite \(K\)-integer

• denominator – a non-zero integral element of \(K\) (default: 1)

EXAMPLES:

sage: Zhat = ProfiniteIntegers(ZZ)
sage: Qhat = ProfiniteNumbers(QQ)
sage: ProfiniteNumber(Qhat, Zhat(2, 15), 8)
1/4 mod 15/8
sage: ProfiniteNumber(Qhat, -97)
-97
sage: ProfiniteNumber(Qhat, Zhat(2, 16), 8)
1/4 mod 2

sage: K.<a> = NumberField(x^4+7)
sage: Ohat = ProfiniteIntegers(K)
sage: Khat = ProfiniteNumbers(K)
sage: ProfiniteNumber(Khat, Ohat(-a^3, a^2+1), 5)
-1/5*a^3 mod (1/5*a^2 + 1/5)

TESTS:

sage: ProfiniteNumber(Qhat, a, 3)
Traceback (most recent call last):
...
TypeError: numerator should be a profinite integer over the base number field
sage: ProfiniteNumber(Khat, Ohat(a), 0)
Traceback (most recent call last):
...
TypeError: denominator should be a non-zero integral element of the base number field

__getitem__(p)

Return the projection of this profinite number to the field of p-adic numbers

INPUT:

• p – a prime number

Only implemented for profinite numbers over \(\QQ\), as no general implementation of completions at finite places of a number field exists in SageMath at the time of writing.

This method implements the projection \(\hat{\QQ} \to \QQ_p\) extending the natural projection \(\hat{\ZZ} \to \ZZ_p\).

EXAMPLES:

sage: a = Qhat(2/35, 2^6 * 3^5 * 5)
sage: a[2]
2 + 2^2 + 2^4 + O(2^6)
sage: a[3]
1 + 2*3 + O(3^5)
sage: a[5]
5^-1 + 2 + O(5)
sage: a[7]
6*7^-1 + O(7^0)
sage: a[11]
O(11^0)

_add_(other)

Return the sum of this profinite number and other

The sum of two profinite numbers \(a\) and \(b\) is defined to be the profinite number \(c\) with smallest represented subset (with respect to inclusion) containing the sum of the represented subsets of \(a\) and \(b\), i.e. containig \(\alpha + \beta\) for all \(\alpha\) represented by \(a\) and \(\beta\) represented by \(b\).

EXAMPLES:

sage: Qhat(Zhat(1, 10), 2) + Qhat(Zhat(2, 15), 3)
7/6 mod 5
sage: Qhat(2, 5) + Qhat(1, 10/3)
4/3 mod 5/3

sage: K.<a> = NumberField(x^3-2)
sage: Ohat = ProfiniteIntegers(K)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(Ohat(a, 10), 3) + Khat(Ohat(a^2, 15), 3)
1/3*a^2 + 1/3*a mod (5/3)
sage: Khat(Ohat(a+1, 20*a+3), 7) + 5
-3749/7*a^2 mod (-20/7*a - 3/7)

REFERENCES:

Section Arithmetic in Section 3.2 of [Her2021].

_sub_(other)

Return the difference of this profinite number and other

The difference of two profinite numbers \(a\) and \(b\) is defined to be the profinite number \(c\) with smallest represented subset (with respect to inclusion) containing the difference of the represented subsets of \(a\) and \(b\), i.e. containig \(\alpha - \beta\) for all \(\alpha\) represented by \(a\) and \(\beta\) represented by \(b\).

EXAMPLES:

sage: Qhat(Zhat(4, 12), 3) - 2/3
2/3 mod 4
sage: 1 - Qhat(Zhat(3, 6), 2)
5/2 mod 3
sage: Qhat(Zhat(4, 12), 3) - Qhat(Zhat(3, 6), 2)
5/6 mod 1

sage: K.<a> = NumberField(x^3-2)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(a, 100) - Khat(a, 20)
0 mod (20)
sage: Khat(a, 100) - Khat(2, 5*a)
a - 2 mod (5*a)

REFERENCES:

Section Arithmetic in Section 3.2 of [Her2021].

_mul_(other)

Return the product of this profinite number and other

The product of two profinite numbers \(a\) and \(b\) is defined to be the profinite number \(c\) with smallest represented subset (with respect to inclusion) containing the product of the represented subsets of \(a\) and \(b\), i.e. containig \(lpha eta\) for all \(lpha\) represented by \(a\) and \(eta\) represented by \(b\).

EXAMPLES:

sage: 1/2 * Qhat(Zhat(5, 15), 2)
5/4 mod 15/4
sage: Qhat(1/3, 30) * Qhat(2, 15)
2/3 mod 5

sage: K.<zeta5> = CyclotomicField(5)
sage: Khat = ProfiniteNumbers(K)
sage: zeta5^3 * Khat(2*zeta5^2, 100/3)
2 mod (100/3*zeta5^3)
sage: Khat(zeta5, 9) * Khat(1/2, 9*zeta5)
1/2*zeta5 mod (9/2)

REFERENCES:

Section Arithmetic in Section 3.2 of [Her2021].

_div_(other)

Return the quotient of this profinite number by other

The profinite number other must have zero-modulus and non-zero value. In other words, it must be the profinite number correspoding to a non-zero element of the base maximal order \(O\).

The quotient of a profinite number \(a\) by an element \(b \in O\) is the unique profinite number whose represented subset equals \(R(a)/b\), where \(R(a)\) denotes the represented subset of \(a\).

EXAMPLES:

sage: Qhat(3/2, 5) / 2
3/4 mod 5/2
sage: Qhat(4, 8) / Qhat(2/3, 0)
6 mod 12

sage: K.<zeta3> = CyclotomicField(3)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(1, 100) / zeta3^2
zeta3 mod (100*zeta3)

REFERENCES:

Section Arithmetic in Section 3.2 of [Her2021].

TESTS:

sage: Khat(1, 100) / Khat(1, 5)
Traceback (most recent call last):
...
ValueError: division by profinite number with non-zero modulus
sage: Khat(1, 100) / Khat(0, 0)
Traceback (most recent call last):
...
ZeroDivisionError: profinite number division by zero

_richcmp_(other, op)

Compare self and other based on the relation op

We only implement equality and non-equality.

We declare a/b equal to c/d if and only if a*d == b*c.

EXAMPLES:

sage: Qhat(Zhat(1, 6), 2) == Qhat(Zhat(2, 12), 4)
True
sage: Qhat(Zhat(1, 6), 2) == Qhat(Zhat(2, 12), 3)
False
sage: Qhat(Zhat(1, 6), 2) != Qhat(Zhat(2, 12), 3)
True

sage: K.<a> = NumberField(x^2+5)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(2, 11/3) == Khat(2, 11)
True
sage: Khat(1, 11/3) == Khat(2, 11)
False

Warning

Equality is not transitive:

sage: a, b, c = Khat(2/7, 1), Khat(1/7, 1/7), Khat(1/7, 1/5)
sage: a == b
True
sage: b == c
True
sage: a == c
False

REFERENCES:

Section 5.4 of [Her2021].

denominator()

Return the denominator of this profinite number

EXAMPLES:

sage: Qhat(1/3, 7/5).denominator()
15
sage: Qhat(Zhat(2, 10), 4).denominator()
2

sage: K.<a> = NumberField(x^2+10)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(a/2, 10/3).denominator()
6

is_integral()

Return whether or not this profinite number is integral

A profinite number is integral if its denominator equals one.

EXAMPLES:

sage: Qhat(7, 11/3).is_integral()
False
sage: Qhat(Zhat(4, 10), 2).is_integral()
True

sage: K.<a> = NumberField(x^2+3)
sage: Khat = ProfiniteNumbers(K)
sage: Khat((a+1)/2, 10).is_integral()
True
sage: Khat((a+1)/4, 10).is_integral()
False

modulus()

Return the modulus of this profinite integer

Denote our base number field by \(K\). If \(K\) is \(\QQ\), then the modulus is a non-negative rational number. Otherwise the modulus is a fractional \(O\)-ideal in \(K\), with \(O\) the maximal order of \(K\).

EXAMPLES:

sage: Qhat(1, 79/90).modulus()
79/90
sage: Qhat(Zhat(4, 10), 7).modulus()
10/7

sage: K.<a> = NumberField(x^2+15)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(3, 4*a).modulus()
Fractional ideal (4*a)
sage: Khat(-1, K.prime_above(2)).modulus()
Fractional ideal (2, 1/2*a - 1/2)

numerator()

Return the numerator of this profinite number

EXAMPLES:

sage: Qhat(1/3, 7/5).numerator()
5 mod 21
sage: Qhat(Zhat(2, 10), 4).numerator()
1 mod 5

sage: K.<a> = NumberField(x^2+10)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(3/2, 10*a).numerator()
3 mod (20*a)
sage: Khat(a/2, 10/3).numerator()
3*a mod (20)

represents(element)

Return wether or not this profinite number represents element

For \(x\) the value of this profinite number and \(m\) its modulus, this profinite number has represented subset \(x + m \hat{O}\), for \(\hat{O}\) the ring of profinite integers over our base number field. So this profinite integer represents \(\alpha\) if and only if \(\alpha \in x + m \hat{O}\).

INPUT:

• element – an element of our base number field

EXAMPLES:

sage: X = [m/n for m in srange(-30, 30) for n in srange(1, 30)]
sage: X = sorted(set(X))
sage: a = Qhat(1/2, 5)
sage: [x for x in X if a.represents(x)]
[-29/2, -19/2, -9/2, 1/2, 11/2, 21/2]
sage: b = Qhat(1/3, 7/2)
sage: [x for x in X if b.represents(x)]
[-20/3, -19/6, 1/3, 23/6, 22/3]

sage: K.<a> = NumberField(x^2-7)
sage: Khat = ProfiniteNumbers(K)
sage: b = Khat(a/2, 10/2)
sage: Y = [m/n for m in srange(-6, 6) for n in srange(1, 6)]
sage: Y = sorted(set(Y))
sage: [y*a+z for y in Y for z in Y if b.represents(y*a+z)] # long time
[1/2*a - 5, 1/2*a, 1/2*a + 5]

to_rational_vector()

Convert this profinite number to a vector of profinite \(\QQ\)-numbers

Let \(K\) be our base number field. This method requires K.gen() to be integral.

Denote K.gen() by \(\alpha\) and K.degree() by \(n\). Then \(\{1, \alpha, \alpha^2, ..., \alpha^{n-1}\}\) is a \(\hat{\QQ}\)-basis of \(\hat{K}\). Hence there is a map \(\phi: \hat{K} \to \hat{\QQ}^n\) that maps \(x \in \hat{K}\) to the unique \((x_0, x_1, ..., x_{n-1}) \in \hat{\QQ}^n\) such that \(x = \sum_{i=0}^{n-1} x_i \alpha^i\). This method implements \(\phi\).

OUTPUT:

A vector \((x_0, ..., x_{n-1})\) of profinite \(\QQ\)-integers such that the following holds. Write \(\phi_i: \hat{K} \to \hat{\QQ}\) for \(\phi\) composed with the projection to the \(i\)-th \(\hat{\QQ}\). Then for any \(\alpha\) that self represents, \(\phi_i(x)\) is represented by \(x_i\).

EXAMPLES:

sage: K.<a> = NumberField(x^2-3)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(1/9 + 7*a).to_rational_vector()
(1/9, 7)
sage: b = Khat(a, 6)
sage: b0, b1 = b.to_rational_vector(); b0, b1
(0 mod 6, 1 mod 6)
sage: b.represents(a)
True
sage: b0.represents(a.vector()[0]) and b1.represents(a.vector()[1])
True
sage: b.represents(a+6)
True
sage: b0.represents((a+6).vector()[0]) and b1.represents((a+6).vector()[1])
True

sage: K.<a> = NumberField(x^3-7)
sage: Khat = ProfiniteNumbers(K)
sage: b = Khat(a^2-1/3, 3*a); b
a^2 - 1/3 mod (3*a)
sage: c = b.to_rational_vector(); c
(8/3 mod 3, 0 mod 3, 1 mod 3)
sage: b.represents(a^2-1/3)
True
sage: all([c[i].represents((a^2-1/3).vector()[i]) for i in range(3)])
True
sage: b.represents(-2*a^2-1/3)
True
sage: all([c[i].represents((-2*a^2-1/3).vector()[i]) for i in range(3)])
True

Base field \(\QQ\) is allowed as well:

sage: Qhat(1/2, 10).to_rational_vector()
(1/2 mod 10,)

REFERENCES:

Section 4.4 of [Her2021].

TESTS:

Check for a previous bug with \([O:\ZZ[\alpha]] > 1\):

sage: K.<a> = NumberField(x^2-5)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(1, 2).to_rational_vector()
(0 mod 1, 0 mod 1)

value()

Return the value of this profinite number

EXAMPLES:

sage: Qhat(7/9, 12).value()
7/9
sage: Qhat(Zhat(5, 10), 2).value()
5/2

sage: K.<a> = NumberField(x^2+15)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(8*a, 30/7).value()
-4/7*a
sage: 8*a - -4/7*a in K.ideal(30/7)
True

class adeles.profinite_number.ProfiniteNumbers(K)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.rings.ring.CommutativeAlgebra

Ring of Profinite Number over a Number Field

_element_constructor_(x, y=None)

Construct a profinite number

INPUT:

Either a value/modulus pair:

• x – an element of our base number field; the value

• y – a fractional ideal of our base number field (default: the zero ideal); the modulus

or a numerator/denominator pair:

• x – a profinite integer over our base number field; the numerator

• y – a non-zero integral element of our base number field (default: 1); the denominator

If the base number field is \(\QQ\), then we also accept a list of \(p\)-adic numbers, for distinct prime numbers \(p\). See _from_padic_numbers() for details.

EXAMPLES:

First we create some profinite numbers by value/modulus pairs:

sage: Qhat(1/3, 10)
1/3 mod 10
sage: Qhat(1000, 7/2)
5/2 mod 7/2
sage: Qhat(9/7)
9/7

sage: K.<a> = NumberField(x^3+x+1)
sage: Khat = ProfiniteNumbers(K)
sage: Khat(a^2, 7/3*(a+1))
a^2 mod (7/3*a + 7/3)
sage: Khat(a)
a

Now we construct some profinite numbers by numerator/denominator pairs:

sage: Qhat(Zhat(1, 10), 2)
1/2 mod 5
sage: Qhat(Zhat(9, 17))
9 mod 17

sage: Khat(Ohat(a, 6), 2)
1/2*a mod (3)
sage: Khat(Ohat(2, 4*a))
2 mod (4*a)

For \(L/K\) a field extension, profinite \(K\)-numbers can be converted to profinite \(L\)-numbers:

sage: R.<t> = K[]
sage: L.<b> = K.extension(t^2-a)
sage: Lhat = ProfiniteNumbers(L)
sage: Lhat(Khat(1/2*a, 10))
1/2*a mod (10)

Lastly, we can also give a list of \(p\)-adic numbers to Qhat:

sage: a_2 = Qp(2)(5/8, 1); a_2
2^-3 + 2^-1 + O(2)
sage: a_3 = Qp(3)(2, 2); a_3
2 + O(3^2)
sage: a_5 = Qp(5)(7/125, -1); a_5
2*5^-3 + 5^-2 + O(5^-1)
sage: a = Qhat([a_2, a_3, a_5]); a
2081/1000 mod 18/5
sage: a[2], a[3], a[5], a[7]
(2^-3 + 2^-1 + O(2), 2 + O(3^2), 2*5^-3 + 5^-2 + O(5^-1), O(7^0))

TESTS:

sage: Qhat('bla')
Traceback (most recent call last):
...
TypeError: Can't construct profinite number from ('bla', None)

_from_padic_numbers(padics)

Construct a profinite \(\QQ\)-number from the list of \(p\)-adic numbers padics

Only implemented over \(\QQ\), as no general implementation of \(p\)-adic numbers currently exists for general number fields.

INPUT:

• padics – a list of \(p\)-adic numbers, for distinct prime numbers \(p\)

OUTPUT:

The unique profinite \(\QQ\) number a such that

• for each prime number p for which a p-adic number a_p exists in padics, we have that a[p] precisely equals a_p;

• for all other prime numbers p, we have that a[p] precisely equals O(p^0).

EXAMPLES:

sage: a_2 = Qp(2)(3/4, 2); a_2
2^-2 + 2^-1 + O(2^2)
sage: a_3 = Qp(3)(1, 1); a_3
1 + O(3)
sage: a = Qhat._from_padic_numbers([a_2, a_3]); a
19/4 mod 12
sage: a[2]
2^-2 + 2^-1 + O(2^2)
sage: a[3]
1 + O(3)
sage: a[5]
O(5^0)

sage: b_2 = Qp(2)(1/6, 1)
sage: b_3 = Qp(3)(3/2, 2)
sage: b = Qhat._from_padic_numbers([b_3, b_2])
sage: b[2] == b_2 and b[3] == b_3
True

TESTS:

sage: c_2 = Qp(2)(2^-25)
sage: c_5 = Qp(5)(0, -3)
sage: c_7 = Qp(7)(2*7^2, 3)
sage: c_41 = Qp(41)(11/41, 8)
sage: c = Qhat._from_padic_numbers([c_41, c_2, c_7, c_5])
sage: c[2] == c_2 and c[5] == c_5 and c[7] == c_7 and c[41] == c_41
True

sage: Qhat._from_padic_numbers([c_2, c_5, c_7, c_2])
Traceback (most recent call last):
...
ValueError: multiple 2-adic integers in profinite integer initialization

Element

alias of ProfiniteNumber

characteristic()

Return the characteristic of this ring, which is zero

EXAMPLES:

sage: Qhat = ProfiniteNumbers()
sage: Qhat.characteristic()
0

construction()

Return a pair (functor, parent) such that functor(parent) returns this profinite integers ring.

EXAMPLES:

sage: F, P = Qhat.construction(); F, P
(ProfiniteCompletionFunctor, Rational Field)
sage: F(P) is Qhat
True

sage: K.<a> = NumberField(x^7-2)
sage: Khat = ProfiniteNumbers(K)
sage: F, P = Khat.construction(); F, P
(ProfiniteCompletionFunctor,
 Number Field in a with defining polynomial x^7 - 2)
sage: F(P) is Khat
True

epsilon()

Return the precision error of elements in this ring

As this depends on the elements, we can give no reasonable answer and hence raise a NotImplementedError.

EXAMPLES:

sage: Qhat.epsilon()
Traceback (most recent call last):
...
NotImplementedError: precision error depends on the elements involved

gen(n=0)

Return the n-th generator of this ring

As this ring has only one generator, we only accept n == 0.

INPUT:

• n – the index of the generator to return (default: 0); must be zero

EXAMPLES:

sage: Qhat.gen()
1
sage: Qhat.gen(0)
1

TESTS:

sage: Qhat.gen(1)
Traceback (most recent call last):
...
IndexError: n must be 0

gens()

Return a tuple of generators of this ring, which is (1,)

EXAMPLES:

sage: Qhat.gens()
(1,)

is_exact()

Return False, indicating that doing arithmetic can lead to precision loss

EXAMPLES:

sage: Qhat.is_exact()
False

is_field(proof=True)

Return False, indicating that this ring is not a field

EXAMPLES:

sage: Qhat.is_field()
False

is_finite()

Return False, indicating that this ring is not finite

EXAMPLES:

sage: Qhat.is_finite()
False

is_integral_domain(proof=None)

Return False, indicating that this ring is not an integral domain

EXAMPLES:

sage: Qhat.is_integral_domain()
False

ngens()

Return the number of generators of this ring, which is \(1\)

EXAMPLES:

sage: Qhat.ngens()
1

number_field()

Return the base number field of this ring of profinite numbers

EXAMPLES:

sage: K.<a> = NumberField(x^5+7*x+9)
sage: Khat = ProfiniteNumbers(K)
sage: Khat.number_field()
Number Field in a with defining polynomial x^5 + 7*x + 9

order()

Return Infinity, indicating that this ring has infinitely many elements

EXAMPLES:

sage: Qhat.order()
+Infinity

random_element()

Return a random profinite number

EXAMPLES:

sage: [Qhat.random_element() for i in range(10)] # random
[0 mod 1/2,
 0 mod 4,
 1/2 mod 2,
 3 mod 7/2,
 1/12 mod 1/6,
 0,
 0 mod 2/9,
 1 mod 2,
 5 mod 6,
 1/2 mod 3/4]

sage: K.<a> = NumberField(x^3-3)
sage: Khat = ProfiniteNumbers(K)
sage: [Khat.random_element() for i in range(10)] # random
[-a^2 - 7*a - 3 mod (22*a - 22),
 1/3*a^2 + 1/3*a + 1 mod (7/3*a^2 - 7/3),
 0 mod (1/2),
 0 mod (1),
 0 mod (1/2*a - 1/2),
 -1/2*a^2 + a - 1/2 mod (5*a^2 + 10*a + 5),
 -18993*a^2 + 1 mod (26*a^2 + 80*a + 70),
 -5*a^2 + a mod (2*a^2 + 3*a - 3),
 9/5*a^2 - 1/5 mod (3/5*a^2 - 3/5),
 -167*a^2 + a + 1 mod (2*a^2 + 8*a - 10)]

some_elements()

Return some elements of this ring

EXAMPLES:

sage: Qhat.some_elements()
[0, 2/5 mod 6/5, 1, 1/3 mod 100]