Profinite Integers¶

Profinite Integers of Number Fields

This file implements rings of profinite integers over number fields and their elements.

Let \(K\) be a number field and \(O\) its ring of integers. For \(O\)-ideals \(I\) and \(J\) with \(I\) dividing \(J\) we have a canonical map \(O/J \to O/I\). The familie \(\{O/I\}_I\) with \(I\) running over the non-zero \(O\)-ideals together with these canonical maps form a projective system of topological rings (where each \(O/I\) is given the discrete topology). Taking the projective limit results in the topological ring

\[\hat{O} = \varprojlim_I O/I\]

of profinite `K`-integers. It is naturally a commutative \(O\)-algebra as well.

Profinite Integers over \(\QQ\)¶

In SageMath the ring of profinite \(\QQ\)-integers \(\hat{\ZZ}\) and its elements look like this:

sage: Zhat
Ring of Profinite Integers of Rational Field
sage: Zhat is ProfiniteIntegers(QQ)
True
sage: Zhat.an_element()
2 mod 6

This element 2 mod 6 is an approximation to the integer 2 in \(\hat{\ZZ}\). It represents all profinite integers in the open subset \(2 + 6\hat{\ZZ}\) of \(\hat{\ZZ}\). Hence 2 mod 6 also represents the integers 8 and -4, but not 5:

sage: a = Zhat(2, 6); a
2 mod 6
sage: a.represents(8)
True
sage: a.represents(-4)
True
sage: a.represents(5)
False
sage: [n for n in range(-20, 30) if a.represents(n)]
[-16, -10, -4, 2, 8, 14, 20, 26]

A sum of a profinite integer in \(2+6\hat{\ZZ}\) and a profinite integer in \(5+15\hat{\ZZ}\) will lie in \(1+3\hat{\ZZ}\), while any product will lie in \(10+30\hat{\ZZ}\):

sage: b = Zhat(5, 15); b
5 mod 15
sage: a + b
1 mod 3
sage: a * b
10 mod 30

The Chinese Remainder Theorem induces a natural isomorphism

\[\hat{\ZZ} \to \prod_p \ZZ_p\]

where \(p\) ranges over all prime numbers and \(\ZZ_p\) denotes the ring of \(p\)-adic integers. This point of view can be taken in SageMath as well:

sage: a_2 = Zp(2)(7, 4); a_2
1 + 2 + 2^2 + O(2^4)
sage: a_5 = Zp(5)(2, 2); a_5
2 + O(5^2)
sage: a = Zhat([a_2, a_5]); a
327 mod 400
sage: a[2]
1 + 2 + 2^2 + O(2^4)
sage: a[3]
O(3^0)
sage: a[5]
2 + O(5^2)
sage: print(a.str(style='padic'))
Profinite integer with values:
  at 2: 1 + 2 + 2^2 + O(2^4)
  at 5: 2 + O(5^2)

As SageMath currently has no implementation of completions of number fields at finite places, the above \(p\)-adic functionality only works for profinite \(\QQ\)-integers.

The general case¶

In general, for \(K\) a number field with ring of integers \(O\), a ProfiniteInteger consists of an element \(x \in O\), called its value, and an ideal \(I\) of \(O\), called its modulus. Such a ProfiniteInteger represents all elements in its represented subset \(x + I \hat{O}\), similar to how the RealInterval RIF(1.2, 1.3) represents all elements of the subset \([1.2, 1,3]\) of \(\RR\).

sage: K.<a> = NumberField(x^2+5)
sage: Ohat = ProfiniteIntegers(K); Ohat
Ring of Profinite Integers of Number Field in a with defining polynomial x^2 + 5
sage: b = Ohat(a+1, K.ideal(4, 2*a+2)); b
a + 1 mod (4, 2*a + 2)
sage: b.value()
a + 1
sage: b.modulus()
Fractional ideal (4, 2*a + 2)

We always keep the value HNF-reduced (cf. Algorithm 1.4.12 of [Coh2000]):

sage: c = Ohat(100000, K.ideal(3, a+1)); c
-a mod (3, a + 1)
sage: c.value()
-a

If a is a ProfiniteInteger representing a profinite \(K\)-integer \(\alpha \in \hat{O}\) and b is a ProfiniteInteger representing \(\beta \in \hat{O}\), then a + b represents \(\alpha + \beta\). Similar statements holds for the other arithmetic operations.

sage: 3*b + c
-a mod (3, a + 1)
sage: b * c
0 mod (2, a + 1)

See the arithmetic methods such as _add_() for details.

In the rest of this file, we shall write “profinite integer” to mean an instance of ProfiniteInteger, as opposed to element of \(\hat{O}\).

See also

profinite_number

Note

Upon creating a number field in SageMath as follows:

sage: K.<a> = NumberField(x^2+5)

the element a has K as its parent, not the maximal order of K. This can cause arithmetic in a ring of profinite integers to give results in a ring of profinite numbers when the user might not expect it:

sage: Ohat = ProfiniteIntegers(K)
sage: b = Ohat(a, 15) + a; b
2*a mod (15)
sage: b.parent() is Ohat
False
sage: from profinite_number import ProfiniteNumbers
sage: b.parent() is ProfiniteNumbers(K)
True

Although above the element a lies in the maximal order of K, its parent is K. Hence the coercion model will look for a common parent of Ohat(a, 15) and a, which is not Ohat, but the ring of profinite numbers over K.

REFERENCES:

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

This implementation of profinite integers 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_integer.ProfiniteCompletionFunctor(args=None, kwds=None)¶

Bases: sage.categories.pushout.ConstructionFunctor

The functor sending (the maximal order of) a number field to its profinite completion

Due to this functor, we have the following functionality.

sage: Zhat(3, 6) + 1/2
7/2 mod 6

sage: R.<X> = ZZ['X']
sage: f = X^2 + 1
sage: f + Zhat(5, 20)
X^2 + 6 mod 20

class adeles.profinite_integer.ProfiniteInteger(parent, value, modulus)¶

Bases: sage.structure.element.CommutativeAlgebraElement

Profinite Integer over a Number Field

REFERENCES:

Section 3.1 of [Her2021].

__init__(parent, value, modulus)¶

Construct the profinite integer “value mod modulus”

INPUT:

parent – a ProfiniteIntegers object for some base number field \(K\)
value – an integral element of \(K\)
modulus – if \(K\) is \(\QQ\): an integer; else: an ideal of \(O\), the maximal order of \(K\)

OUTPUT:

The profinite integer representing the subset value + modulus * \(\hat{O}\) of \(\hat{O}\), the profinite completion of \(O\).

EXAMPLES:

sage: ProfiniteInteger(Zhat, 4, 15)
4 mod 15

sage: K.<a> = NumberField(x^7+3)
sage: Ohat = ProfiniteIntegers(K)
sage: ProfiniteInteger(Ohat, a^6-a, 30*a^3)
a^6 - a mod (30*a^3)
sage: I = K.ideal(a^5+3*a^2, 120)
sage: ProfiniteInteger(Ohat, 7, I)
14*a^6 + 1 mod (-a^6 + a^5 + 3*a - 3)

TESTS:

sage: Zhat(3, -10)
3 mod 10
sage: ProfiniteInteger(Ohat, a^2, 0)
a^2
sage: ProfiniteInteger(Ohat, None, a)
Traceback (most recent call last):
...
TypeError: value must be an element of Maximal Order in Number Field in a with defining polynomial x^7 + 3
sage: ProfiniteInteger(Ohat, a, None)
Traceback (most recent call last):
...
TypeError: modulus must be an ideal of Maximal Order in Number Field in a with defining polynomial x^7 + 3
sage: ProfiniteInteger(Ohat, a, 1/a)
Traceback (most recent call last):
...
TypeError: modulus must be an ideal of Maximal Order in Number Field in a with defining polynomial x^7 + 3
sage: ProfiniteInteger(Zhat, 3, 1/25)
Traceback (most recent call last):
...
TypeError: modulus must be an integer

__getitem__(p)¶

Return the projection of this profinite integer to the ring of p-adic integers

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

The Chinese Remainder Theorem induces a canonical isomorphism \(\hat{\ZZ} \cong \prod_p \ZZ_p\), where \(p\) runs over all prime numbers. This method computes the image of self under this isomorphism and returns the projection to the p-th coordinate.

INPUT:

p – a prime number

EXAMPLES:

sage: a = Zhat(4, 27)
sage: a[3]
1 + 3 + O(3^3)
sage: b = Zhat(5, 2^6 * 5^12)
sage: b[2]
1 + 2^2 + O(2^6)
sage: b[3]
O(3^0)
sage: b[5]
5 + O(5^12)
sage: c = Zhat(-1, 7^5)
sage: c[7]
6 + 6*7 + 6*7^2 + 6*7^3 + 6*7^4 + O(7^5)
sage: c[7].parent()
7-adic Ring with capped relative precision 20

TESTS:

sage: d[2]
Traceback (most recent call last):
...
NotImplementedError: projection to `p`-adics only implemented over rationals
sage: d[K.prime_above(5)]
Traceback (most recent call last):
...
NotImplementedError: projection to `p`-adics only implemented over rationals
sage: c[-1]
Traceback (most recent call last):
...
ValueError: p must be a prime number
sage: c[6]
Traceback (most recent call last):
...
ValueError: p must be a prime number

_add_(other)¶

Return the sum of this profinite integer and other

The sum of two profinite integers \(a\) and \(b\) is defined to be the profinite integer \(c\) 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: Zhat = ProfiniteIntegers(ZZ)
sage: a = Zhat(3, 20)
sage: b = Zhat(-1, 30)
sage: a + b
2 mod 10
sage: a + 3
6 mod 20

sage: K.<a> = NumberField(x^2-7)
sage: O = K.maximal_order()
sage: Ohat = ProfiniteIntegers(O)
sage: b = Ohat(3*a+1, a*13)
sage: c = Ohat(2*a+1, 7)
sage: b+c
2 mod (a)
sage: O(-a)+c
a + 1 mod (7)

REFERENCES:

Section Arithmetic of Section 3.1 of [Her2021].

_sub_(other)¶

Return the difference of this profinite integer and other

The difference of two profinite integers \(a\) and \(b\) is defined to be the profinite integer 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: Zhat = ProfiniteIntegers(ZZ)
sage: a = Zhat(3, 20)
sage: b = Zhat(-1, 30)
sage: a-b
4 mod 10
sage: b-0
29 mod 30

sage: K.<a> = NumberField(x^2-7)
sage: Ohat = ProfiniteIntegers(K)
sage: b = Ohat(3*a+1, a*13)
sage: c = Ohat(2*a+1, 7)
sage: b-c
0 mod (a)
sage: c-1
2*a mod (7)

REFERENCES:

Section Arithmetic of Section 3.1 of [Her2021].

_mul_(other)¶

Return the product of this profinite integer and other

The product of two profinite integers \(a\) and \(b\) is defined to be the profinite integer 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: Zhat = ProfiniteIntegers(ZZ)
sage: a = Zhat(3, 20)
sage: b = Zhat(-1, 30)
sage: a*b
7 mod 10
sage: 0*a
0

sage: K.<a> = NumberField(x^2-7)
sage: Ohat = ProfiniteIntegers(K)
sage: b = Ohat(3*a+1, a*13)
sage: c = Ohat(2*a+1, 7)
sage: b*c
1 mod (a)
sage: 1*c
2*a + 1 mod (7)

REFERENCES:

Section Arithmetic of Section 3.1 of [Her2021].

_div_(other)¶

Divide this profinite integer by other in the ring of profinite numbers and return the result

Only implemented for other having zero-modulus and non-zero value.

EXAMPLES:

sage: b = Zhat(4, 10) / 7; b
4/7 mod 10/7
sage: b.parent()
Profinite Numbers of Rational Field

sage: K.<a> = NumberField(x^2+x-7)
sage: Ohat = ProfiniteIntegers(K)
sage: Ohat(a^2, 20) / a
a mod (20/7*a + 20/7)

TESTS:

sage: Zhat(4, 10) / Zhat(2, 10)
Traceback (most recent call last):
...
NotImplementedError: Cannot divide by profinite integers of non-zero modulus

_floordiv_(other)¶

Return the quotient of this profinite integer by other

Only implemented when other has zero modulus and self.value() and self.modulus() are both divisible by other.value().

EXAMPLES:

sage: Zhat(5, 100) // 5
1 mod 20

sage: K.<a> = NumberField(x^2+5)
sage: Ohat = ProfiniteIntegers(K)
sage: Ohat(9*a, 75) // O(3*a)
3 mod (-5*a)

TESTS:

sage: b // Zhat(5, 10)
Traceback (most recent call last):
...
TypeError: can only divide by elements of the base
sage: b/0
Traceback (most recent call last):
...
ZeroDivisionError: division by zero
sage: b/3
Traceback (most recent call last):
...
NotImplementedError: value is not divisible by 3
sage: c/9
Traceback (most recent call last):
...
NotImplementedError: modulus is not divisible by 9

_richcmp_(other, op)¶

Compare this profinite integer to other based on the relation op

We only implement equality and non-equality.

Two elements x mod I and y mod J are considered equal if and only if \(x \equiv y \mod (I+J)\).

EXAMPLES:

sage: K.<a> = NumberField(x^4+17)
sage: O = K.maximal_order()
sage: Ohat = ProfiniteIntegers(O)
sage: b = O(a^2+1)
sage: c = Ohat(a^2+1, 1000)
sage: d = Ohat(a^2+11, a*10)
sage: e = Ohat(a^3+a-5, 170)
sage: c == c
True

Transitivity of equality does not hold:

sage: b == c
True
sage: c == d
True
sage: b == d
False

sage: c == e
False
sage: c != e
True
sage: b != d
True
sage: c != d
False

REFERENCES:

Section 5.4 of [Her2021].

TESTS:

sage: b < d
Traceback (most recent call last):
...
NotImplementedError: only equality and inequality are implemented

factorial_digits()¶

Return the factorial digits of this profinite integer

Only implemented for profinite integers over \(\QQ\) with non-zero modulus.

Let \(k\) be the largest integer such that \(k!\) divides our modulus (this is called our “factorial precision”). Then our factorial digits are the unique integers \(d_1, d_2, ..., d_{k-1}\) satisfying \(0 \leq d_i \leq i\) such that

\[x \equiv d_1 \cdot 1! + d_2 \cdot 2! + ... + d_{k-1} \cdot (k-1)! \mod k!\]

where \(x\) denotes the value of this profinite integer.

EXAMPLES:

sage: digits = Zhat(11, 48).factorial_digits(); digits
[1, 2, 1]
sage: sum([digits[i] * factorial(i+1) for i in range(3)])
11

REFERENCES:

Section 7.2 of [Her2021].

is_integral()¶

Return True, indicating that this profinite integer is integral

EXAMPLES:

sage: Zhat(-5, 12).is_integral()
True

is_unit()¶

Return whether or not self could be a unit

More precisely, we return True if and only if the subset of profinite integers that self represents contains a unit.

If self is x mod m, this is equivalent to \(x \in (O/mO)^*\), where \(O\) denotes our base ring of integers.

EXAMPLES:

sage: Zhat(3, 0).is_unit()
False
sage: Zhat(-1, 0).is_unit()
True
sage: Zhat(3, 25).is_unit()
True
sage: Zhat(5, 25).is_unit()
False

sage: K.<a> = NumberField(x^3-2)
sage: Ohat = ProfiniteIntegers(K)
sage: Ohat(3, 8).is_unit()
True
sage: Ohat(a, 8).is_unit()
False

modulus()¶

Return the modulus of this profinite integer

This is an integer if the base number field is QQ. Otherwise it is an ideal of the base maximal order.

EXAMPLES:

sage: Zhat(50, 100).modulus()
100

sage: K.<a> = NumberField(x^4-17)
sage: Ohat = ProfiniteIntegers(K)
sage: Ohat(a^3, a*160).modulus()
Fractional ideal (2720, 160*a)

represents(element)¶

Return whether or not this profinite integer represents element

The represented subset of this profinite integer is \(x + m \hat{O}\), where \(x\) is our value, \(m\) is our modulus and \(\hat{O}\) is the ring of profinite numbers over our base field. Hence this profinite number represents \(\alpha \in \hat{O}\) if and only if \(\alpha \in x + m \hat{O}\).

INPUT:

element – an integral element of the base number field

EXAMPLES:

sage: b = Zhat(3, 10)
sage: [n for n in range(-50, 50) if b.represents(n)]
[-47, -37, -27, -17, -7, 3, 13, 23, 33, 43]

sage: K.<a> = NumberField(x^2+2)
sage: Ohat = ProfiniteIntegers(K)
sage: c = Ohat(a+1, 3*a)
sage: [m*a+n for m in range(-7, 7) for n in range(-7, 7) if c.represents(m*a+n)]
[-5*a - 5, -5*a + 1, -2*a - 5, -2*a + 1, a - 5, a + 1, 4*a - 5, 4*a + 1]

str(style='value-modulus')¶

Return a string representation of this profinite integer

INPUT:

style – string (default: “value-modulus”); style to use. The other options are “factorial” and “padic”.

EXAMPLES:

sage: a = Zhat(970, 2^7 * 3^3 * 5^2)
sage: print(a.str(style='value-modulus'))
970 mod 86400
sage: print(a.str(style='factorial'))
2*2! + 1*3! + 2*5! + O(6!)
sage: print(a.str(style='padic'))
Profinite integer with values:
  at 2: 2 + 2^3 + 2^6 + O(2^7)
  at 3: 1 + 2*3 + 2*3^2 + O(3^3)
  at 5: 4*5 + O(5^2)

TESTS:

sage: print(a.str(style='blah'))
Traceback (most recent call last):
...
ValueError: unkown style ``blah''

value()¶

Return the value of this profinite integer

EXAMPLES:

sage: Zhat(3, 6).value()
3
sage: Zhat(7, 6).value() # the value is reduced
1

visual()¶

Return the smallest closed interval within the unitinterval \([0,1]\) in which the image of this profinite integer under the visualization map is contained

The visualization function is defined by

\[\phi(a) = \sum_{i=1}^\infty \frac{d_i}{(1+i)!}\]

for \(a \in \hat{\ZZ}\) with factorial digit sequence \((d_i)_{i=1}^\infty\).

EXAMPLES:

The subset \(1 + 2\hat{\ZZ}\) of \(\hat{\ZZ}\) maps onto \([1/2, 1]\) by \(\phi\). Hence we have

sage: Zhat(1, 2).visual()
(1/2, 1)

The subset \(2 + 3\hat{\ZZ}\) is mapped onto \([1/6, 1/3] \cup [5/6, 1]\) by \(\phi\). So we get

sage: Zhat(2, 3).visual()
(1/6, 1)

REFERENCES:

Section 7.3 of [Her2021].

class adeles.profinite_integer.ProfiniteIntegers(O)¶

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

Ring of profinite integers over a number field

REFERENCES:

Section 3.1 of [Her2021].

_element_constructor_(x, y=None)¶

Construct a profinite integer

INPUT:

We accept many input formats. The most common one is the following:

x – element of the base maximal order; the value
y – ideal of the base maximal order; the modulus

All other formats only accept one argument, which can be one of the following:

a profinite \(K\)-integer for \(K\) a subfield of our base number field
an integral profinite \(K\)-number for \(K\) a subfield of our base number field
an element of a quotient of our base maximal order
an element of our base maximal order

For constructing profinite \(\QQ\)-integers, even more formats are accepted, namely:

a factorial digit list, i.e. a list of non-negative integers such that the \(i\)-th entry is at most \(i+1\) (see factorial_digits()).
a \(p\)-adic integer, for some prime number \(p\)
a list of \(p\)-adic integers for distinct prime numbers \(p\)

EXAMPLES:

We start with standard (value, modulus) input:

sage: Zhat(-4, 17)
13 mod 17
sage: K.<a> = NumberField(x^3-5*x^2+1)
sage: Ohat = ProfiniteIntegers(K)
sage: Ohat(a^2+1, 4*a)
a^2 + 1 mod (4*a)

Upon giving a ProfiniteInteger as input we get:

sage: Zhat(Zhat(3, 5))
3 mod 5
sage: Ohat(Zhat(4, 6))
-2 mod (6)
sage: Ohat(Ohat(a, 8))
a mod (8)

Integral profinite numbers can be converted to profinite integers:

sage: Zhat(Qhat(-1, 5))
4 mod 5
sage: Khat = ProfiniteNumbers(K)
sage: Ohat(Khat(a+1, 6))
a + 1 mod (6)

Quotient ring elements can be given as input as well:

sage: Zhat(Zmod(40)(7))
7 mod 40
sage: R = K.maximal_order().quotient(30*a, 'b')
sage: Ohat(R(2*a+3))
2*a + 3 mod (30*a)

Elements of the base maximal order a coerced to profinite integers with zero modulus:

sage: Zhat(97)
97
sage: Ohat(79*a)
79*a

Profinite integers over \(\QQ\) can be constructed from factorial digits:

sage: Zhat([1, 0, 0, 2, 0])
49 mod 720

And profinite integers over \(\QQ\) can also be constructed from \(p\)-adic integers:

sage: Zhat(Zp(5)(-1))
95367431640624 mod 95367431640625
sage: Zhat([Zp(2)(20, 5), Zp(3)(7, 2)])
52 mod 288

Element¶: alias of ProfiniteInteger

characteristic()¶

Return the characteristic of this ring, which is zero

EXAMPLES:

sage: Zhat = ProfiniteIntegers()
sage: Zhat.characteristic()
0

construction()¶

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

EXAMPLES:

sage: F, P = Zhat.construction(); F, P
(ProfiniteCompletionFunctor, Integer Ring)
sage: F(P) is Zhat
True

sage: K.<a> = NumberField(x^3-2)
sage: Ohat = ProfiniteIntegers(K)
sage: F, P = Ohat.construction(); F, P
(ProfiniteCompletionFunctor,
 Maximal Order in Number Field in a with defining polynomial x^3 - 2)
sage: F(P) is Ohat
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: Zhat.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: Zhat.gen()
1
sage: Zhat.gen(0)
1

TESTS:

sage: Zhat.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: Zhat.gens()
(1,)

is_exact()¶

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

EXAMPLES:

sage: Zhat.is_exact()
False

is_field(proof=True)¶

Return False, indicating that this ring is not a field

Note that this ring of profinite integers is (canonically isomorphic to) the product of all completions of \(K\) at the finite primes of \(K\), where \(K\) denotes our base number field. As a product of non-trivial rings, this ring is clearly not a field.

EXAMPLES:

sage: Zhat.is_field()
False

is_finite()¶

Return False, indicating that this ring is not finite

EXAMPLES:

sage: Zhat.is_finite()
False

is_integral_domain(proof=None)¶

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

Note that this ring of profinite integers is (canonically isomorphic to) the product of all completions of \(K\) at the finite primes of \(K\), where \(K\) denotes our base number field. As a product of non-trivial rings, this ring is clearly not an integral domain.

EXAMPLES:

sage: Zhat = ProfiniteIntegers()
sage: Zhat.is_integral_domain()
False

is_noetherian()¶

Return False, indicating that this ring is not Noetherian

Note that this ring of profinite integers is (canonically isomorphic to) the product of all completions of \(K\) at the finite primes of \(K\), where \(K\) denotes our base number field. As a product of infinitely many non-trivial rings, this ring is clearly not Noetherian.

EXAMPLES:

sage: Zhat.is_noetherian()
False

krull_dimension()¶

Return 1, indiciting that the Krull dimension of this ring is one

Note that this ring of profinite integers is (canonically isomorphic to) the product of all completions of \(K\) at the finite primes of \(K\), where \(K\) denotes our base number field. Each such completion has Krull dimension one and therefore this ring has Krull dimension one as well.

EXAMPLES:

sage: Zhat.krull_dimension()
1

ngens()¶

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

EXAMPLES:

sage: Zhat.ngens()
1

number_field()¶

Return the base number field of this ring of profinite integers

The base number field equals the fraction field of self.base().

EXAMPLES:

sage: Zhat = ProfiniteIntegers()
sage: Zhat.number_field()
Rational Field
sage: K.<a> = NumberField(x^2+x-7)
sage: Ohat = ProfiniteIntegers(K)
sage: Ohat.number_field()
Number Field in a with defining polynomial x^2 + x - 7

order()¶

Return Infinity, indicating that this ring has infinitely many elements

EXAMPLES:

sage: Zhat.order()
+Infinity

random_element()¶

Return a random element of this ring

EXAMPLES:

sage: Ohat.random_element() # random
0 mod (1)
sage: Ohat.random_element() # random
2/3*a^2 mod (a - 1)
sage: Ohat.random_element() # random
1/3*a^2 + a mod (-1/3*a^2 + a - 1)
sage: Ohat.random_element() # random
4/3*a^2 + a mod (-10773*a^2 - 75978*a - 229392)
sage: Ohat.random_element() # random
0 mod (1)
sage: Ohat.random_element() # random
a^2 + 3*a - 1 mod (104*a^2 + 52*a + 364)

some_elements()¶

Return some elements of this ring

EXAMPLES:

sage: Zhat.some_elements()
[0, 2 mod 6, 1, 66 mod 79, 20, 10 mod 100]

Profinite Integers¶

Profinite Integers over \(\QQ\)¶

The general case¶

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