Embeddings into ambient fields¶
This module provides classes to handle embeddings of number fields into ambient fields (generally \(\RR\) or \(\CC\)).
- class sage.rings.number_field.number_field_morphisms.CyclotomicFieldConversion[source]¶
Bases:
Map
This allows one to cast one cyclotomic field in another consistently.
EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import CyclotomicFieldConversion sage: K1.<z1> = CyclotomicField(12) sage: K2.<z2> = CyclotomicField(18) sage: f = CyclotomicFieldConversion(K1, K2) sage: f(z1^2) z2^3 sage: f(z1) Traceback (most recent call last): ... ValueError: Element z1 has no image in the codomain
>>> from sage.all import * >>> from sage.rings.number_field.number_field_morphisms import CyclotomicFieldConversion >>> K1 = CyclotomicField(Integer(12), names=('z1',)); (z1,) = K1._first_ngens(1) >>> K2 = CyclotomicField(Integer(18), names=('z2',)); (z2,) = K2._first_ngens(1) >>> f = CyclotomicFieldConversion(K1, K2) >>> f(z1**Integer(2)) z2^3 >>> f(z1) Traceback (most recent call last): ... ValueError: Element z1 has no image in the codomain
Tests from Issue #29511:
sage: K.<z> = CyclotomicField(12) sage: K1.<z1> = CyclotomicField(3) sage: K(2) in K1 # indirect doctest True sage: K1(K(2)) # indirect doctest 2
>>> from sage.all import * >>> K = CyclotomicField(Integer(12), names=('z',)); (z,) = K._first_ngens(1) >>> K1 = CyclotomicField(Integer(3), names=('z1',)); (z1,) = K1._first_ngens(1) >>> K(Integer(2)) in K1 # indirect doctest True >>> K1(K(Integer(2))) # indirect doctest 2
- class sage.rings.number_field.number_field_morphisms.CyclotomicFieldEmbedding[source]¶
Bases:
NumberFieldEmbedding
Specialized class for converting cyclotomic field elements into a cyclotomic field of higher order. All the real work is done by
_lift_cyclotomic_element()
.- section()[source]¶
Return the section of
self
.EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import CyclotomicFieldEmbedding sage: K = CyclotomicField(7) sage: L = CyclotomicField(21) sage: f = CyclotomicFieldEmbedding(K, L) sage: h = f.section() sage: h(f(K.gen())) # indirect doctest zeta7
>>> from sage.all import * >>> from sage.rings.number_field.number_field_morphisms import CyclotomicFieldEmbedding >>> K = CyclotomicField(Integer(7)) >>> L = CyclotomicField(Integer(21)) >>> f = CyclotomicFieldEmbedding(K, L) >>> h = f.section() >>> h(f(K.gen())) # indirect doctest zeta7
- class sage.rings.number_field.number_field_morphisms.EmbeddedNumberFieldConversion[source]¶
Bases:
Map
This allows one to cast one number field in another consistently, assuming they both have specified embeddings into an ambient field (by default it looks for an embedding into \(\CC\)).
This is done by factoring the minimal polynomial of the input in the number field of the codomain. This may fail if the element is not actually in the given field.
- class sage.rings.number_field.number_field_morphisms.EmbeddedNumberFieldMorphism[source]¶
Bases:
NumberFieldEmbedding
This allows one to go from one number field in another consistently, assuming they both have specified embeddings into an ambient field.
If no ambient field is supplied, then the following ambient fields are tried:
the pushout of the fields where the number fields are embedded;
the algebraic closure of the previous pushout;
\(\CC\).
EXAMPLES:
sage: x = polygen(ZZ, 'x') sage: K.<i> = NumberField(x^2 + 1, embedding=QQbar(I)) sage: L.<i> = NumberField(x^2 + 1, embedding=-QQbar(I)) sage: from sage.rings.number_field.number_field_morphisms import EmbeddedNumberFieldMorphism sage: EmbeddedNumberFieldMorphism(K, L, CDF) Generic morphism: From: Number Field in i with defining polynomial x^2 + 1 with i = I To: Number Field in i with defining polynomial x^2 + 1 with i = -I Defn: i -> -i sage: EmbeddedNumberFieldMorphism(K, L, QQbar) Generic morphism: From: Number Field in i with defining polynomial x^2 + 1 with i = I To: Number Field in i with defining polynomial x^2 + 1 with i = -I Defn: i -> -i
>>> from sage.all import * >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(2) + Integer(1), embedding=QQbar(I), names=('i',)); (i,) = K._first_ngens(1) >>> L = NumberField(x**Integer(2) + Integer(1), embedding=-QQbar(I), names=('i',)); (i,) = L._first_ngens(1) >>> from sage.rings.number_field.number_field_morphisms import EmbeddedNumberFieldMorphism >>> EmbeddedNumberFieldMorphism(K, L, CDF) Generic morphism: From: Number Field in i with defining polynomial x^2 + 1 with i = I To: Number Field in i with defining polynomial x^2 + 1 with i = -I Defn: i -> -i >>> EmbeddedNumberFieldMorphism(K, L, QQbar) Generic morphism: From: Number Field in i with defining polynomial x^2 + 1 with i = I To: Number Field in i with defining polynomial x^2 + 1 with i = -I Defn: i -> -i
- section()[source]¶
EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import EmbeddedNumberFieldMorphism sage: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^2 - 700, embedding=25) sage: L.<b> = NumberField(x^6 - 700, embedding=3) sage: f = EmbeddedNumberFieldMorphism(K, L) sage: f(2*a - 1) 2*b^3 - 1 sage: g = f.section() sage: g(2*b^3 - 1) 2*a - 1
>>> from sage.all import * >>> from sage.rings.number_field.number_field_morphisms import EmbeddedNumberFieldMorphism >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(2) - Integer(700), embedding=Integer(25), names=('a',)); (a,) = K._first_ngens(1) >>> L = NumberField(x**Integer(6) - Integer(700), embedding=Integer(3), names=('b',)); (b,) = L._first_ngens(1) >>> f = EmbeddedNumberFieldMorphism(K, L) >>> f(Integer(2)*a - Integer(1)) 2*b^3 - 1 >>> g = f.section() >>> g(Integer(2)*b**Integer(3) - Integer(1)) 2*a - 1
- class sage.rings.number_field.number_field_morphisms.NumberFieldEmbedding[source]¶
Bases:
Morphism
If R is a lazy field, the closest root to gen_embedding will be chosen.
EXAMPLES:
sage: x = polygen(QQ) sage: from sage.rings.number_field.number_field_morphisms import NumberFieldEmbedding sage: K.<a> = NumberField(x^3-2) sage: f = NumberFieldEmbedding(K, RLF, 1) sage: f(a)^3 2.00000000000000? sage: RealField(200)(f(a)^3) 2.0000000000000000000000000000000000000000000000000000000000 sage: sigma_a = K.polynomial().change_ring(CC).roots()[1][0]; sigma_a -0.62996052494743... - 1.09112363597172*I sage: g = NumberFieldEmbedding(K, CC, sigma_a) sage: g(a+1) 0.37003947505256... - 1.09112363597172*I
>>> from sage.all import * >>> x = polygen(QQ) >>> from sage.rings.number_field.number_field_morphisms import NumberFieldEmbedding >>> K = NumberField(x**Integer(3)-Integer(2), names=('a',)); (a,) = K._first_ngens(1) >>> f = NumberFieldEmbedding(K, RLF, Integer(1)) >>> f(a)**Integer(3) 2.00000000000000? >>> RealField(Integer(200))(f(a)**Integer(3)) 2.0000000000000000000000000000000000000000000000000000000000 >>> sigma_a = K.polynomial().change_ring(CC).roots()[Integer(1)][Integer(0)]; sigma_a -0.62996052494743... - 1.09112363597172*I >>> g = NumberFieldEmbedding(K, CC, sigma_a) >>> g(a+Integer(1)) 0.37003947505256... - 1.09112363597172*I
- gen_image()[source]¶
Return the image of the generator under this embedding.
EXAMPLES:
sage: f = QuadraticField(7, 'a', embedding=2).coerce_embedding() sage: f.gen_image() 2.645751311064591?
>>> from sage.all import * >>> f = QuadraticField(Integer(7), 'a', embedding=Integer(2)).coerce_embedding() >>> f.gen_image() 2.645751311064591?
- sage.rings.number_field.number_field_morphisms.closest(target, values, margin=1)[source]¶
This is a utility function that returns the item in
values
closest to target (with respect to theabs
function). Ifmargin
is greater than 1, and \(x\) and \(y\) are the first and second closest elements totarget
, then only return \(x\) if \(x\) ismargin
times closer totarget
than \(y\), i.e.margin * abs(target-x) < abs(target-y)
.
- sage.rings.number_field.number_field_morphisms.create_embedding_from_approx(K, gen_image)[source]¶
Return an embedding of
K
determined bygen_image
.The codomain of the embedding is the parent of
gen_image
or, ifgen_image
is not already an exact root of the defining polynomial ofK
, the corresponding lazy field. The embedding maps the generator ofK
to a root of the defining polynomial ofK
closest togen_image
.EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import create_embedding_from_approx sage: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^3 - x + 1/10) sage: create_embedding_from_approx(K, 1) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Real Lazy Field Defn: a -> 0.9456492739235915? sage: create_embedding_from_approx(K, 0) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Real Lazy Field Defn: a -> 0.10103125788101081? sage: create_embedding_from_approx(K, -1) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Real Lazy Field Defn: a -> -1.046680531804603?
>>> from sage.all import * >>> from sage.rings.number_field.number_field_morphisms import create_embedding_from_approx >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(3) - x + Integer(1)/Integer(10), names=('a',)); (a,) = K._first_ngens(1) >>> create_embedding_from_approx(K, Integer(1)) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Real Lazy Field Defn: a -> 0.9456492739235915? >>> create_embedding_from_approx(K, Integer(0)) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Real Lazy Field Defn: a -> 0.10103125788101081? >>> create_embedding_from_approx(K, -Integer(1)) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Real Lazy Field Defn: a -> -1.046680531804603?
We can define embeddings from one number field to another:
sage: L.<b> = NumberField(x^6-x^2+1/10) sage: create_embedding_from_approx(K, b^2) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Number Field in b with defining polynomial x^6 - x^2 + 1/10 Defn: a -> b^2
>>> from sage.all import * >>> L = NumberField(x**Integer(6)-x**Integer(2)+Integer(1)/Integer(10), names=('b',)); (b,) = L._first_ngens(1) >>> create_embedding_from_approx(K, b**Integer(2)) Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 1/10 To: Number Field in b with defining polynomial x^6 - x^2 + 1/10 Defn: a -> b^2
If the embedding is exact, it must be valid:
sage: create_embedding_from_approx(K, b) Traceback (most recent call last): ... ValueError: b is not a root of x^3 - x + 1/10
>>> from sage.all import * >>> create_embedding_from_approx(K, b) Traceback (most recent call last): ... ValueError: b is not a root of x^3 - x + 1/10
- sage.rings.number_field.number_field_morphisms.matching_root(poly, target, ambient_field=None, margin=1, max_prec=None)[source]¶
Given a polynomial and a
target
, choose the root thattarget
best approximates as compared inambient_field
.If the parent of
target
is exact, the equality is required, otherwise find closest root (with respect to theabs
function) in the ambient field to thetarget
, and return the root ofpoly
(if any) that approximates it best.EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import matching_root sage: R.<x> = CC[] sage: matching_root(x^2-2, 1.5) 1.41421356237310 sage: matching_root(x^2-2, -100.0) -1.41421356237310 sage: matching_root(x^2-2, .00000001) 1.41421356237310 sage: matching_root(x^3-1, CDF.0) -0.50000000000000... + 0.86602540378443...*I sage: matching_root(x^3-x, 2, ambient_field=RR) 1.00000000000000
>>> from sage.all import * >>> from sage.rings.number_field.number_field_morphisms import matching_root >>> R = CC['x']; (x,) = R._first_ngens(1) >>> matching_root(x**Integer(2)-Integer(2), RealNumber('1.5')) 1.41421356237310 >>> matching_root(x**Integer(2)-Integer(2), -RealNumber('100.0')) -1.41421356237310 >>> matching_root(x**Integer(2)-Integer(2), RealNumber('.00000001')) 1.41421356237310 >>> matching_root(x**Integer(3)-Integer(1), CDF.gen(0)) -0.50000000000000... + 0.86602540378443...*I >>> matching_root(x**Integer(3)-x, Integer(2), ambient_field=RR) 1.00000000000000
- sage.rings.number_field.number_field_morphisms.root_from_approx(f, a)[source]¶
Return an exact root of the polynomial \(f\) closest to \(a\).
INPUT:
f
– polynomial with rational coefficientsa
– element of a ring
OUTPUT:
A root of
f
in the parent ofa
or, ifa
is not already an exact root off
, in the corresponding lazy field. The root is taken to be closest toa
among all roots off
.EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import root_from_approx sage: R.<x> = QQ[] sage: root_from_approx(x^2 - 1, -1) -1 sage: root_from_approx(x^2 - 2, 1) 1.414213562373095? sage: root_from_approx(x^3 - x - 1, RR(1)) 1.324717957244746? sage: root_from_approx(x^3 - x - 1, CC.gen()) -0.6623589786223730? + 0.5622795120623013?*I sage: root_from_approx(x^2 + 1, 0) Traceback (most recent call last): ... ValueError: x^2 + 1 has no real roots sage: root_from_approx(x^2 + 1, CC(0)) -1*I sage: root_from_approx(x^2 - 2, sqrt(2)) # needs sage.symbolic sqrt(2) sage: root_from_approx(x^2 - 2, sqrt(3)) # needs sage.symbolic Traceback (most recent call last): ... ValueError: sqrt(3) is not a root of x^2 - 2
>>> from sage.all import * >>> from sage.rings.number_field.number_field_morphisms import root_from_approx >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> root_from_approx(x**Integer(2) - Integer(1), -Integer(1)) -1 >>> root_from_approx(x**Integer(2) - Integer(2), Integer(1)) 1.414213562373095? >>> root_from_approx(x**Integer(3) - x - Integer(1), RR(Integer(1))) 1.324717957244746? >>> root_from_approx(x**Integer(3) - x - Integer(1), CC.gen()) -0.6623589786223730? + 0.5622795120623013?*I >>> root_from_approx(x**Integer(2) + Integer(1), Integer(0)) Traceback (most recent call last): ... ValueError: x^2 + 1 has no real roots >>> root_from_approx(x**Integer(2) + Integer(1), CC(Integer(0))) -1*I >>> root_from_approx(x**Integer(2) - Integer(2), sqrt(Integer(2))) # needs sage.symbolic sqrt(2) >>> root_from_approx(x**Integer(2) - Integer(2), sqrt(Integer(3))) # needs sage.symbolic Traceback (most recent call last): ... ValueError: sqrt(3) is not a root of x^2 - 2