Field \(\QQ\) of Rational Numbers

The class RationalField represents the field \(\QQ\) of (arbitrary precision) rational numbers. Each rational number is an instance of the class Rational.

Interactively, an instance of RationalField is available as QQ:

sage: QQ
Rational Field

Values of various types can be converted to rational numbers by using the __call__ method of RationalField (that is, by treating QQ as a function).

sage: RealField(9).pi()
3.1
sage: QQ(RealField(9).pi())
22/7
sage: QQ(RealField().pi())
245850922/78256779
sage: QQ(35)
35
sage: QQ('12/347')
12/347
sage: QQ(exp(pi*I))
-1
sage: x = polygen(ZZ)
sage: QQ((3*x)/(4*x))
3/4

TEST:

sage: Q = RationalField()
sage: Q == loads(dumps(Q))
True
sage: RationalField() is RationalField()
True

AUTHORS:

  • Niles Johnson (2010-08): trac ticket #3893: random_element() should pass on *args and **kwds.
  • Travis Scrimshaw (2012-10-18): Added additional docstrings for full coverage. Removed duplicates of discriminant() and signature().
class sage.rings.rational_field.RationalField

Bases: sage.rings.rational_field._uniq, sage.rings.number_field.number_field_base.NumberField

The class RationalField represents the field \(\QQ\) of rational numbers.

EXAMPLES:

sage: a = long(901824309821093821093812093810928309183091832091)
sage: b = QQ(a); b
901824309821093821093812093810928309183091832091
sage: QQ(b)
901824309821093821093812093810928309183091832091
sage: QQ(int(93820984323))
93820984323
sage: QQ(ZZ(901824309821093821093812093810928309183091832091))
901824309821093821093812093810928309183091832091
sage: QQ('-930482/9320842317')
-930482/9320842317
sage: QQ((-930482, 9320842317))
-930482/9320842317
sage: QQ([9320842317])
9320842317
sage: QQ(pari(39029384023840928309482842098430284398243982394))
39029384023840928309482842098430284398243982394
sage: QQ('sage')
Traceback (most recent call last):
...
TypeError: unable to convert sage to a rational

Coercion from the reals to the rational is done by default using continued fractions.

sage: QQ(RR(3929329/32))
3929329/32
sage: QQ(-RR(3929329/32))
-3929329/32
sage: QQ(RR(1/7)) - 1/7
0

If you specify an optional second base argument, then the string representation of the float is used.

sage: QQ(23.2, 2)
6530219459687219/281474976710656
sage: 6530219459687219.0/281474976710656
23.20000000000000
sage: a = 23.2; a
23.2000000000000
sage: QQ(a, 10)
116/5

Here’s a nice example involving elliptic curves:

sage: E = EllipticCurve('11a')
sage: L = E.lseries().at1(300)[0]; L
0.2538418608559106843377589233...
sage: O = E.period_lattice().omega(); O
1.26920930427955
sage: t = L/O; t
0.200000000000000
sage: QQ(RealField(45)(t))
1/5
absolute_degree()

Return the absolute degree of \(\QQ\) which is 1.

EXAMPLES:

sage: QQ.absolute_degree()
1
absolute_discriminant()

Return the absolute discriminant, which is 1.

EXAMPLES:

sage: QQ.absolute_discriminant()
1
algebraic_closure()

Return the algebraic closure of self (which is \(\QQbar\)).

EXAMPLES:

sage: QQ.algebraic_closure()
Algebraic Field
characteristic()

Return 0 since the rational field has characteristic 0.

EXAMPLES:

sage: c = QQ.characteristic(); c
0
sage: parent(c)
Integer Ring
class_number()

Return the class number of the field of rational numbers, which is 1.

EXAMPLES:

sage: QQ.class_number()
1
completion(p, prec, extras={})

Return the completion of \(\QQ\) at \(p\).

EXAMPLES:

sage: QQ.completion(infinity, 53)
Real Field with 53 bits of precision
sage: QQ.completion(5, 15, {'print_mode': 'bars'})
5-adic Field with capped relative precision 15
complex_embedding(prec=53)

Return embedding of the rational numbers into the complex numbers.

EXAMPLES:

sage: QQ.complex_embedding()
Ring morphism:
  From: Rational Field
  To:   Complex Field with 53 bits of precision
  Defn: 1 |--> 1.00000000000000
sage: QQ.complex_embedding(20)
Ring morphism:
  From: Rational Field
  To:   Complex Field with 20 bits of precision
  Defn: 1 |--> 1.0000
construction()

Returns a pair (functor, parent) such that functor(parent) returns self.

This is the construction of \(\QQ\) as the fraction field of \(\ZZ\).

EXAMPLES:

sage: QQ.construction()
(FractionField, Integer Ring)
degree()

Return the degree of \(\QQ\) which is 1.

EXAMPLES:

sage: QQ.degree()
1
discriminant()

Return the discriminant of the field of rational numbers, which is 1.

EXAMPLES:

sage: QQ.discriminant()
1
embeddings(K)

Return list of the one embedding of \(\QQ\) into \(K\), if it exists.

EXAMPLES:

sage: QQ.embeddings(QQ)
[Ring Coercion endomorphism of Rational Field]
sage: QQ.embeddings(CyclotomicField(5))
[Ring Coercion morphism:
  From: Rational Field
  To:   Cyclotomic Field of order 5 and degree 4]

\(K\) must have characteristic 0:

sage: QQ.embeddings(GF(3))
Traceback (most recent call last):
...
ValueError: no embeddings of the rational field into K.
extension(poly, names, check=True, embedding=None)

Create a field extension of \(\QQ\).

EXAMPLES:

We make a single absolute extension:

sage: K.<a> = QQ.extension(x^3 + 5); K
Number Field in a with defining polynomial x^3 + 5

We make an extension generated by roots of two polynomials:

sage: K.<a,b> = QQ.extension([x^3 + 5, x^2 + 3]); K
Number Field in a with defining polynomial x^3 + 5 over its base field
sage: b^2
-3
sage: a^3
-5
gen(n=0)

Return the n-th generator of \(\QQ\).

There is only the 0-th generator which is 1.

EXAMPLES:

sage: QQ.gen()
1
gens()

Return a tuple of generators of \(\QQ\) which is only (1,).

EXAMPLES:

sage: QQ.gens()
(1,)
is_absolute()

\(\QQ\) is an absolute extension of \(\QQ\).

EXAMPLES:

sage: QQ.is_absolute()
True
is_field(proof=True)

Return True, since the rational field is a field.

EXAMPLES:

sage: QQ.is_field()
True
is_finite()

Return False, since the rational field is not finite.

EXAMPLES:

sage: QQ.is_finite()
False
is_prime_field()

Return True since \(\QQ\) is a prime field.

EXAMPLES:

sage: QQ.is_prime_field()
True
is_subring(K)

Return True if \(\QQ\) is a subring of \(K\).

We are only able to determine this in some cases, e.g., when \(K\) is a field or of positive characteristic.

EXAMPLES:

sage: QQ.is_subring(QQ)
True
sage: QQ.is_subring(QQ['x'])
True
sage: QQ.is_subring(GF(7))
False
sage: QQ.is_subring(CyclotomicField(7))
True
sage: QQ.is_subring(ZZ)
False
sage: QQ.is_subring(Frac(ZZ))
True
maximal_order()

Return the maximal order of the rational numbers, i.e., the ring \(\ZZ\) of integers.

EXAMPLES:

sage: QQ.maximal_order()
Integer Ring
sage: QQ.ring_of_integers ()
Integer Ring
ngens()

Return the number of generators of \(\QQ\) which is 1.

EXAMPLES:

sage: QQ.ngens()
1
number_field()

Return the number field associated to \(\QQ\). Since \(\QQ\) is a number field, this just returns \(\QQ\) again.

EXAMPLES:

sage: QQ.number_field() is QQ
True
order()

Return the order of \(\QQ\) which is \(\infty\).

EXAMPLES:

sage: QQ.order()
+Infinity
places(all_complex=False, prec=None)

Return the collection of all infinite places of self, which in this case is just the embedding of self into \(\RR\).

By default, this returns homomorphisms into RR. If prec is not None, we simply return homomorphisms into RealField(prec) (or RDF if prec=53).

There is an optional flag all_complex, which defaults to False. If all_complex is True, then the real embeddings are returned as embeddings into the corresponding complex field.

For consistency with non-trivial number fields.

EXAMPLES:

sage: QQ.places()
[Ring morphism:
  From: Rational Field
  To:   Real Field with 53 bits of precision
  Defn: 1 |--> 1.00000000000000]
sage: QQ.places(prec=53)
[Ring morphism:
  From: Rational Field
  To:   Real Double Field
  Defn: 1 |--> 1.0]
sage: QQ.places(prec=200, all_complex=True)
[Ring morphism:
  From: Rational Field
  To:   Complex Field with 200 bits of precision
  Defn: 1 |--> 1.0000000000000000000000000000000000000000000000000000000000]
power_basis()

Return a power basis for this number field over its base field.

The power basis is always [1] for the rational field. This method is defined to make the rational field behave more like a number field.

EXAMPLES:

sage: QQ.power_basis()
[1]
primes_of_bounded_norm_iter(B)

Iterator yielding all primes less than or equal to \(B\).

INPUT:

  • B – a positive integer; upper bound on the primes generated.

OUTPUT:

An iterator over all integer primes less than or equal to \(B\).

Note

This function exists for compatibility with the related number field method, though it returns prime integers, not ideals.

EXAMPLES:

sage: it = QQ.primes_of_bounded_norm_iter(10)
sage: list(it)
[2, 3, 5, 7]
sage: list(QQ.primes_of_bounded_norm_iter(1))
[]
random_element(num_bound=None, den_bound=None, *args, **kwds)

Return an random element of \(\QQ\).

EXAMPLES:

sage: QQ.random_element(10,10)
1/4

Passes extra positional or keyword arguments through:

sage: QQ.random_element(10,10, distribution='1/n')
-1
range_by_height(start, end=None)

Range function for rational numbers, ordered by height.

Returns a Python generator for the list of rational numbers with heights in range(start, end). Follows the same convention as Python range, see range? for details.

See also __iter__().

EXAMPLES:

All rational numbers with height strictly less than 4:

sage: list(QQ.range_by_height(4))
[0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3, -3, 2/3, -2/3, 3/2, -3/2]
sage: [a.height() for a in QQ.range_by_height(4)]
[1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3]

All rational numbers with height 2:

sage: list(QQ.range_by_height(2, 3))
[1/2, -1/2, 2, -2]

Nonsensical integer arguments will return an empty generator:

sage: list(QQ.range_by_height(3, 3))
[]
sage: list(QQ.range_by_height(10, 1))
[]

There are no rational numbers with height \(\leq 0\):

sage: list(QQ.range_by_height(-10, 1))
[]
relative_discriminant()

Return the relative discriminant, which is 1.

EXAMPLES:

sage: QQ.relative_discriminant()
1
residue_field(p, check=True)

Return the residue field of \(\QQ\) at the prime \(p\), for consistency with other number fields.

INPUT:

  • p - a prime integer.
  • check (default True) - if True check the primality of \(p\), else do not.

OUTPUT: The residue field at this prime.

EXAMPLES:

sage: QQ.residue_field(5)
Residue field of Integers modulo 5
sage: QQ.residue_field(next_prime(10^9))
Residue field of Integers modulo 1000000007
selmer_group(S, m, proof=True, orders=False)

Compute the group \(\QQ(S,m)\).

INPUT:

  • S – a set of primes
  • m – a positive integer
  • proof – ignored
  • orders (default False) – if True, output two lists, the generators and their orders

OUTPUT:

A list of generators of \(\QQ(S,m)\) (and, optionally, their orders in \(\QQ^\times/(\QQ^\times)^m\)). This is the subgroup of \(\QQ^\times/(\QQ^\times)^m\) consisting of elements \(a\) such that the valuation of \(a\) is divisible by \(m\) at all primes not in \(S\). It is equal to the group of \(S\)-units modulo \(m\)-th powers. The group \(\QQ(S,m)\) contains the subgroup of those \(a\) such that \(\QQ(\sqrt[m]{a})/\QQ\) is unramified at all primes of \(\QQ\) outside of \(S\), but may contain it properly when not all primes dividing \(m\) are in \(S\).

EXAMPLES:

sage: QQ.selmer_group((), 2)
[-1]
sage: QQ.selmer_group((3,), 2)
[-1, 3]
sage: QQ.selmer_group((5,), 2)
[-1, 5]

The previous examples show that the group generated by the output may be strictly larger than the ‘true’ Selmer group of elements giving extensions unramified outside \(S\).

When \(m\) is even, \(-1\) is a generator of order \(2\):

sage: QQ.selmer_group((2,3,5,7,), 2, orders=True)
([-1, 2, 3, 5, 7], [2, 2, 2, 2, 2])
sage: QQ.selmer_group((2,3,5,7,), 3, orders=True)
([2, 3, 5, 7], [3, 3, 3, 3])
selmer_group_iterator(S, m, proof=True)

Return an iterator through elements of the finite group \(\QQ(S,m)\).

INPUT:

  • S – a set of primes
  • m – a positive integer
  • proof – ignored

OUTPUT:

An iterator yielding the distinct elements of \(\QQ(S,m)\). See the docstring for selmer_group() for more information.

EXAMPLES:

sage: list(QQ.selmer_group_iterator((), 2))
[1, -1]
sage: list(QQ.selmer_group_iterator((2,), 2))
[1, 2, -1, -2]
sage: list(QQ.selmer_group_iterator((2,3), 2))
[1, 3, 2, 6, -1, -3, -2, -6]
sage: list(QQ.selmer_group_iterator((5,), 2))
[1, 5, -1, -5]
signature()

Return the signature of the rational field, which is \((1,0)\), since there are 1 real and no complex embeddings.

EXAMPLES:

sage: QQ.signature()
(1, 0)
zeta(n=2)

Return a root of unity in self.

INPUT:

  • n – integer (default: 2) order of the root of unity

EXAMPLES:

sage: QQ.zeta()
-1
sage: QQ.zeta(2)
-1
sage: QQ.zeta(1)
1
sage: QQ.zeta(3)
Traceback (most recent call last):
...
ValueError: no n-th root of unity in rational field
sage.rings.rational_field.frac(n, d)

Return the fraction n/d.

EXAMPLES:

sage: from sage.rings.rational_field import frac
sage: frac(1,2)
1/2
sage.rings.rational_field.is_RationalField(x)

Check to see if x is the rational field.

EXAMPLES:

sage: from sage.rings.rational_field import is_RationalField as is_RF
sage: is_RF(QQ)
True
sage: is_RF(ZZ)
False

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