TESTS:
sage: k = NumberField(x^2 + 1, 'i'); k == loads(dumps(k))
True
Bases: sage.rings.ring.Field
Synomym for self.maximal_order(...).
EXAMPLES:
sage: NumberField(x^3 - 2,'a').OK()
Maximal Order in Number Field in a with defining polynomial x^3 - 2
Return the Bach bound associated to this number field. Assuming the General Riemann Hypothesis, this is a bound B so that every integral ideal is equivalent modulo principal fractional ideals to an integral ideal of norm at most B.
See also
OUTPUT:
symbolic expression or the Integer 1
EXAMPLES:
We compute both the Minkowski and Bach bounds for a quadratic field, where the Minkowski bound is much better:
sage: K = QQ[sqrt(5)]
sage: K.minkowski_bound()
1/2*sqrt(5)
sage: K.minkowski_bound().n()
1.11803398874989
sage: K.bach_bound()
12*log(5)^2
sage: K.bach_bound().n()
31.0834847277628
We compute both the Minkowski and Bach bounds for a bigger degree field, where the Bach bound is much better:
sage: K = CyclotomicField(37)
sage: K.minkowski_bound().n()
7.50857335698544e14
sage: K.bach_bound().n()
191669.304126267
Return the degree of this number field.
EXAMPLES:
sage: NumberField(x^3 + 9, 'a').degree()
3
Return the discriminant of this number field.
EXAMPLES:
sage: NumberField(x^3 + 9, 'a').discriminant()
-243
Return True if self is viewed as a single extension over Q.
EXAMPLES:
sage: K.<a> = NumberField(x^3+2)
sage: K.is_absolute()
True
sage: y = polygen(K)
sage: L.<b> = NumberField(y^2+1)
sage: L.is_absolute()
False
sage: QQ.is_absolute()
True
Return False since number fields are not finite.
EXAMPLES:
sage: z = polygen(QQ)
sage: K.<theta, beta> = NumberField([z^3 - 3, z^2 + 1])
sage: K.is_finite()
False
sage: K.order()
+Infinity
Return the maximal order, i.e., the ring of integers of this number field.
EXAMPLES:
sage: NumberField(x^3 - 2,'b').maximal_order()
Maximal Order in Number Field in b with defining polynomial x^3 - 2
Return the Minkowski bound associated to this number field, which is a bound B so that every integral ideal is equivalent modulo principal fractional ideals to an integral ideal of norm at most B.
See also
OUTPUT:
symbolic expression or Rational
EXAMPLES:
The Minkowski bound for tells us that the class
number is 1:
sage: K = QQ[I]
sage: B = K.minkowski_bound(); B
4/pi
sage: B.n()
1.27323954473516
We compute the Minkowski bound for :
sage: K = QQ[2^(1/3)]
sage: B = K.minkowski_bound(); B
16/3*sqrt(3)/pi
sage: B.n()
2.94042077558289
sage: int(B)
2
We compute the Minkowski bound for , which has class
number 2:
sage: K = QQ[sqrt(10)]
sage: B = K.minkowski_bound(); B
sqrt(10)
sage: int(B)
3
sage: K.class_number()
2
The bound of course also works for the rational numbers:
sage: QQ.minkowski_bound()
1
Synomym for self.maximal_order(...).
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 1)
sage: K.ring_of_integers()
Maximal Order in Number Field in a with defining polynomial x^2 + 1
Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of this field, respectively.
EXAMPLES:
sage: NumberField(x^3 - 2, 'a').signature()
(1, 1)
Return True if x is of number field type.
EXAMPLES:
sage: from sage.rings.number_field.number_field_base import is_NumberField
sage: is_NumberField(NumberField(x^2+1,'a'))
True
sage: is_NumberField(QuadraticField(-97,'theta'))
True
sage: is_NumberField(CyclotomicField(97))
True
Note that the rational numbers QQ are a number field.:
sage: is_NumberField(QQ)
True
sage: is_NumberField(ZZ)
False