Finite residue fields#

We can take the residue field of maximal ideals in the ring of integers of number fields. We can also take the residue field of irreducible polynomials over \(\GF{p}\).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)
sage: P = K.ideal(29).factor()[0][0]
sage: k = K.residue_field(P); k
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
sage: k.order()
841

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]
>>> k = K.residue_field(P); k
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
>>> k.order()
841

We reduce mod a prime for which the ring of integers is not monogenic (i.e., 2 is an essential discriminant divisor):

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: F = K.factor(2); F
(Fractional ideal (-1/2*a^2 + 1/2*a - 1)) * (Fractional ideal (-a^2 + 2*a - 3))
* (Fractional ideal (3/2*a^2 - 5/2*a + 4))
sage: F[0][0].residue_field()
Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F[1][0].residue_field()
Residue field of Fractional ideal (-a^2 + 2*a - 3)
sage: F[2][0].residue_field()
Residue field of Fractional ideal (3/2*a^2 - 5/2*a + 4)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), names=('a',)); (a,) = K._first_ngens(1)
>>> F = K.factor(Integer(2)); F
(Fractional ideal (-1/2*a^2 + 1/2*a - 1)) * (Fractional ideal (-a^2 + 2*a - 3))
* (Fractional ideal (3/2*a^2 - 5/2*a + 4))
>>> F[Integer(0)][Integer(0)].residue_field()
Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
>>> F[Integer(1)][Integer(0)].residue_field()
Residue field of Fractional ideal (-a^2 + 2*a - 3)
>>> F[Integer(2)][Integer(0)].residue_field()
Residue field of Fractional ideal (3/2*a^2 - 5/2*a + 4)

We can also form residue fields from \(\ZZ\):

sage: ZZ.residue_field(17)
Residue field of Integers modulo 17

>>> from sage.all import *
>>> ZZ.residue_field(Integer(17))
Residue field of Integers modulo 17

And for polynomial rings over finite fields:

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(5)[]
sage: I = R.ideal(t^2 + 2)
sage: k = ResidueField(I); k
Residue field in tbar of Principal ideal (t^2 + 2) of
 Univariate Polynomial Ring in t over Finite Field of size 5

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(5))['t']; (t,) = R._first_ngens(1)
>>> I = R.ideal(t**Integer(2) + Integer(2))
>>> k = ResidueField(I); k
Residue field in tbar of Principal ideal (t^2 + 2) of
 Univariate Polynomial Ring in t over Finite Field of size 5

AUTHORS:

  • David Roe (2007-10-3): initial version

  • William Stein (2007-12): bug fixes

  • John Cremona (2008-9): extend reduction maps to the whole valuation ring add support for residue fields of ZZ

  • David Roe (2009-12): added support for \(GF(p)(t)\) and moved to new coercion framework.

class sage.rings.finite_rings.residue_field.LiftingMap[source]#

Bases: Section

Lifting map from residue class field to number field.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + 2)
sage: F = K.factor(5)[0][0].residue_field()
sage: F.degree()
2
sage: L = F.lift_map(); L
Lifting map:
  From: Residue field in abar of Fractional ideal (a^2 + 2*a - 1)
  To:   Maximal Order generated by a in Number Field in a with defining polynomial x^3 + 2
sage: L(F.0^2)
3*a + 1
sage: L(3*a + 1) == F.0^2
True

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(13)[]
sage: P = R.ideal(8*t^12 + 9*t^11 + 11*t^10 + 2*t^9 + 11*t^8
....:             + 3*t^7 + 12*t^6 + t^4 + 7*t^3 + 5*t^2 + 12*t + 1)
sage: k.<a> = P.residue_field()
sage: k.lift_map()
Lifting map:
  From: Residue field in a of Principal ideal (t^12 + 6*t^11 + 3*t^10
        + 10*t^9 + 3*t^8 + 2*t^7 + 8*t^6 + 5*t^4 + 9*t^3 + 12*t^2 + 8*t + 5) of
        Univariate Polynomial Ring in t over Finite Field of size 13
  To:   Univariate Polynomial Ring in t over Finite Field of size 13

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> F = K.factor(Integer(5))[Integer(0)][Integer(0)].residue_field()
>>> F.degree()
2
>>> L = F.lift_map(); L
Lifting map:
  From: Residue field in abar of Fractional ideal (a^2 + 2*a - 1)
  To:   Maximal Order generated by a in Number Field in a with defining polynomial x^3 + 2
>>> L(F.gen(0)**Integer(2))
3*a + 1
>>> L(Integer(3)*a + Integer(1)) == F.gen(0)**Integer(2)
True

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(13))['t']; (t,) = R._first_ngens(1)
>>> P = R.ideal(Integer(8)*t**Integer(12) + Integer(9)*t**Integer(11) + Integer(11)*t**Integer(10) + Integer(2)*t**Integer(9) + Integer(11)*t**Integer(8)
...             + Integer(3)*t**Integer(7) + Integer(12)*t**Integer(6) + t**Integer(4) + Integer(7)*t**Integer(3) + Integer(5)*t**Integer(2) + Integer(12)*t + Integer(1))
>>> k = P.residue_field(names=('a',)); (a,) = k._first_ngens(1)
>>> k.lift_map()
Lifting map:
  From: Residue field in a of Principal ideal (t^12 + 6*t^11 + 3*t^10
        + 10*t^9 + 3*t^8 + 2*t^7 + 8*t^6 + 5*t^4 + 9*t^3 + 12*t^2 + 8*t + 5) of
        Univariate Polynomial Ring in t over Finite Field of size 13
  To:   Univariate Polynomial Ring in t over Finite Field of size 13
class sage.rings.finite_rings.residue_field.ReductionMap[source]#

Bases: Map

A reduction map from a (subset) of a number field or function field to this residue class field.

It will be defined on those elements of the field with non-negative valuation at the specified prime.

EXAMPLES:

sage: # needs sage.rings.number_field sage.symbolic
sage: I = QQ[sqrt(17)].factor(5)[0][0]; I
Fractional ideal (5)
sage: k = I.residue_field(); k
Residue field in sqrt17bar of Fractional ideal (5)
sage: R = k.reduction_map(); R
Partially defined reduction map:
  From: Number Field in sqrt17 with defining polynomial x^2 - 17
        with sqrt17 = 4.123105625617660?
  To:   Residue field in sqrt17bar of Fractional ideal (5)

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(next_prime(2^20))[]; P = R.ideal(t^2 + t + 1)
sage: k = P.residue_field()
sage: k.reduction_map()
Partially defined reduction map:
  From: Fraction Field of
        Univariate Polynomial Ring in t over Finite Field of size 1048583
  To:   Residue field in tbar of Principal ideal (t^2 + t + 1) of
        Univariate Polynomial Ring in t over Finite Field of size 1048583

>>> from sage.all import *
>>> # needs sage.rings.number_field sage.symbolic
>>> I = QQ[sqrt(Integer(17))].factor(Integer(5))[Integer(0)][Integer(0)]; I
Fractional ideal (5)
>>> k = I.residue_field(); k
Residue field in sqrt17bar of Fractional ideal (5)
>>> R = k.reduction_map(); R
Partially defined reduction map:
  From: Number Field in sqrt17 with defining polynomial x^2 - 17
        with sqrt17 = 4.123105625617660?
  To:   Residue field in sqrt17bar of Fractional ideal (5)

>>> # needs sage.rings.finite_rings
>>> R = GF(next_prime(Integer(2)**Integer(20)))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(2) + t + Integer(1))
>>> k = P.residue_field()
>>> k.reduction_map()
Partially defined reduction map:
  From: Fraction Field of
        Univariate Polynomial Ring in t over Finite Field of size 1048583
  To:   Residue field in tbar of Principal ideal (t^2 + t + 1) of
        Univariate Polynomial Ring in t over Finite Field of size 1048583
section()[source]#

Computes a section of the map, namely a map that lifts elements of the residue field to elements of the field.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^5 - 5*x + 2)
sage: P = K.ideal(47).factor()[0][0]
sage: k = K.residue_field(P)
sage: f = k.convert_map_from(K)
sage: s = f.section(); s
Lifting map:
  From: Residue field in abar of
        Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
  To:   Number Field in a with defining polynomial x^5 - 5*x + 2
sage: s(k.gen())
a
sage: L.<b> = NumberField(x^5 + 17*x + 1)
sage: P = L.factor(53)[0][0]
sage: l = L.residue_field(P)
sage: g = l.convert_map_from(L)
sage: s = g.section(); s
Lifting map:
  From: Residue field in bbar of Fractional ideal (53, b^2 + 23*b + 8)
  To:   Number Field in b with defining polynomial x^5 + 17*x + 1
sage: s(l.gen()).parent()
Number Field in b with defining polynomial x^5 + 17*x + 1

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(2)[]; h = t^5 + t^2 + 1
sage: k.<a> = R.residue_field(h)
sage: K = R.fraction_field()
sage: f = k.convert_map_from(K)
sage: f.section()                                                           # needs sage.libs.ntl
Lifting map:
  From: Residue field in a of Principal ideal (t^5 + t^2 + 1) of
        Univariate Polynomial Ring in t over Finite Field of size 2 (using GF2X)
  To:   Fraction Field of
        Univariate Polynomial Ring in t over Finite Field of size 2 (using GF2X)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(5) - Integer(5)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(47)).factor()[Integer(0)][Integer(0)]
>>> k = K.residue_field(P)
>>> f = k.convert_map_from(K)
>>> s = f.section(); s
Lifting map:
  From: Residue field in abar of
        Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
  To:   Number Field in a with defining polynomial x^5 - 5*x + 2
>>> s(k.gen())
a
>>> L = NumberField(x**Integer(5) + Integer(17)*x + Integer(1), names=('b',)); (b,) = L._first_ngens(1)
>>> P = L.factor(Integer(53))[Integer(0)][Integer(0)]
>>> l = L.residue_field(P)
>>> g = l.convert_map_from(L)
>>> s = g.section(); s
Lifting map:
  From: Residue field in bbar of Fractional ideal (53, b^2 + 23*b + 8)
  To:   Number Field in b with defining polynomial x^5 + 17*x + 1
>>> s(l.gen()).parent()
Number Field in b with defining polynomial x^5 + 17*x + 1

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(2))['t']; (t,) = R._first_ngens(1); h = t**Integer(5) + t**Integer(2) + Integer(1)
>>> k = R.residue_field(h, names=('a',)); (a,) = k._first_ngens(1)
>>> K = R.fraction_field()
>>> f = k.convert_map_from(K)
>>> f.section()                                                           # needs sage.libs.ntl
Lifting map:
  From: Residue field in a of Principal ideal (t^5 + t^2 + 1) of
        Univariate Polynomial Ring in t over Finite Field of size 2 (using GF2X)
  To:   Fraction Field of
        Univariate Polynomial Ring in t over Finite Field of size 2 (using GF2X)
class sage.rings.finite_rings.residue_field.ResidueFieldFactory[source]#

Bases: UniqueFactory

A factory that returns the residue class field of a prime ideal \(p\) of the ring of integers of a number field, or of a polynomial ring over a finite field.

INPUT:

  • p – a prime ideal of an order in a number field.

  • names – the variable name for the finite field created. Defaults to the name of the number field variable but with bar placed after it.

  • check – whether or not to check if \(p\) is prime.

OUTPUT:

The residue field at the prime \(p\).

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)                                              # needs sage.rings.number_field
sage: P = K.ideal(29).factor()[0][0]                                            # needs sage.rings.number_field
sage: ResidueField(P)                                                           # needs sage.rings.number_field
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]                                            # needs sage.rings.number_field
>>> ResidueField(P)                                                           # needs sage.rings.number_field
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)

The result is cached:

sage: ResidueField(P) is ResidueField(P)                                        # needs sage.rings.number_field
True
sage: k = K.residue_field(P); k                                                 # needs sage.rings.number_field
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
sage: k.order()                                                                 # needs sage.rings.number_field
841

>>> from sage.all import *
>>> ResidueField(P) is ResidueField(P)                                        # needs sage.rings.number_field
True
>>> k = K.residue_field(P); k                                                 # needs sage.rings.number_field
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
>>> k.order()                                                                 # needs sage.rings.number_field
841

It also works for polynomial rings:

sage: R.<t> = GF(31)[]
sage: P = R.ideal(t^5 + 2*t + 11)
sage: ResidueField(P)                                                           # needs sage.rings.finite_rings
Residue field in tbar of Principal ideal (t^5 + 2*t + 11) of
 Univariate Polynomial Ring in t over Finite Field of size 31

sage: ResidueField(P) is ResidueField(P)                                        # needs sage.rings.finite_rings
True
sage: k = ResidueField(P); k.order()                                            # needs sage.rings.finite_rings
28629151

>>> from sage.all import *
>>> R = GF(Integer(31))['t']; (t,) = R._first_ngens(1)
>>> P = R.ideal(t**Integer(5) + Integer(2)*t + Integer(11))
>>> ResidueField(P)                                                           # needs sage.rings.finite_rings
Residue field in tbar of Principal ideal (t^5 + 2*t + 11) of
 Univariate Polynomial Ring in t over Finite Field of size 31

>>> ResidueField(P) is ResidueField(P)                                        # needs sage.rings.finite_rings
True
>>> k = ResidueField(P); k.order()                                            # needs sage.rings.finite_rings
28629151

An example where the generator of the number field doesn’t generate the residue class field:

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 - 875)
sage: P = K.ideal(5).factor()[0][0]; k = K.residue_field(P); k
Residue field in abar of Fractional ideal (5, 1/25*a^2 - 2/5*a - 1)
sage: k.polynomial()
abar^2 + 3*abar + 4
sage: k.0^3 - 875
2

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(x**Integer(3) - Integer(875), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(5)).factor()[Integer(0)][Integer(0)]; k = K.residue_field(P); k
Residue field in abar of Fractional ideal (5, 1/25*a^2 - 2/5*a - 1)
>>> k.polynomial()
abar^2 + 3*abar + 4
>>> k.gen(0)**Integer(3) - Integer(875)
2

An example where the residue class field is large but of degree 1:

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 - 875)
sage: P = K.ideal(2007).factor()[2][0]; k = K.residue_field(P); k
Residue field of Fractional ideal (223, 1/5*a + 11)
sage: k(a)
168
sage: k(a)^3 - 875
0

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(x**Integer(3) - Integer(875), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(2007)).factor()[Integer(2)][Integer(0)]; k = K.residue_field(P); k
Residue field of Fractional ideal (223, 1/5*a + 11)
>>> k(a)
168
>>> k(a)**Integer(3) - Integer(875)
0

And for polynomial rings:

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(next_prime(2^18))[]
sage: P = R.ideal(t - 5)
sage: k = ResidueField(P); k
Residue field of Principal ideal (t + 262142) of
 Univariate Polynomial Ring in t over Finite Field of size 262147
sage: k(t)
5

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = GF(next_prime(Integer(2)**Integer(18)))['t']; (t,) = R._first_ngens(1)
>>> P = R.ideal(t - Integer(5))
>>> k = ResidueField(P); k
Residue field of Principal ideal (t + 262142) of
 Univariate Polynomial Ring in t over Finite Field of size 262147
>>> k(t)
5

In this example, 2 is an inessential discriminant divisor, so divides the index of ZZ[a] in the maximal order for all a:

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: P = K.ideal(2).factor()[0][0]; P
Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F = K.residue_field(P); F
Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F(a)
0
sage: B = K.maximal_order().basis(); B
[1, 1/2*a^2 + 1/2*a, a^2]
sage: F(B[1])
1
sage: F(B[2])
0
sage: F
Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F.degree()
1

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(2)).factor()[Integer(0)][Integer(0)]; P
Fractional ideal (-1/2*a^2 + 1/2*a - 1)
>>> F = K.residue_field(P); F
Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
>>> F(a)
0
>>> B = K.maximal_order().basis(); B
[1, 1/2*a^2 + 1/2*a, a^2]
>>> F(B[Integer(1)])
1
>>> F(B[Integer(2)])
0
>>> F
Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
>>> F.degree()
1
create_key_and_extra_args(p, names=None, check=True, impl=None, **kwds)[source]#

Return a tuple containing the key (uniquely defining data) and any extra arguments.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)                                          # needs sage.rings.number_field
sage: ResidueField(K.ideal(29).factor()[0][0])  # indirect doctest          # needs sage.rings.number_field
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> ResidueField(K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)])  # indirect doctest          # needs sage.rings.number_field
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
create_object(version, key, **kwds)[source]#

Create the object from the key and extra arguments. This is only called if the object was not found in the cache.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)                                          # needs sage.rings.number_field
sage: P = K.ideal(29).factor()[0][0]                                        # needs sage.rings.number_field
sage: ResidueField(P) is ResidueField(P)  # indirect doctest                # needs sage.rings.number_field
True

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]                                        # needs sage.rings.number_field
>>> ResidueField(P) is ResidueField(P)  # indirect doctest                # needs sage.rings.number_field
True
class sage.rings.finite_rings.residue_field.ResidueFieldHomomorphism_global[source]#

Bases: RingHomomorphism

The class representing a homomorphism from the order of a number field or function field to the residue field at a given prime.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)
sage: P  = K.ideal(29).factor()[0][0]
sage: k  = K.residue_field(P)
sage: OK = K.maximal_order()
sage: abar = k(OK.1); abar
abar
sage: (1+abar)^179
24*abar + 12

sage: # needs sage.rings.number_field
sage: phi = k.coerce_map_from(OK); phi
Ring morphism:
  From: Maximal Order generated by a in Number Field in a with defining polynomial x^3 - 7
  To:   Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
sage: phi in Hom(OK,k)
True
sage: phi(OK.1)
abar

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(19)[]; P = R.ideal(t^2 + 5)
sage: k.<a> = R.residue_field(P)
sage: f = k.coerce_map_from(R); f
Ring morphism:
  From: Univariate Polynomial Ring in t over Finite Field of size 19
  To:   Residue field in a of Principal ideal (t^2 + 5) of
        Univariate Polynomial Ring in t over Finite Field of size 19

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P  = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]
>>> k  = K.residue_field(P)
>>> OK = K.maximal_order()
>>> abar = k(OK.gen(1)); abar
abar
>>> (Integer(1)+abar)**Integer(179)
24*abar + 12

>>> # needs sage.rings.number_field
>>> phi = k.coerce_map_from(OK); phi
Ring morphism:
  From: Maximal Order generated by a in Number Field in a with defining polynomial x^3 - 7
  To:   Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
>>> phi in Hom(OK,k)
True
>>> phi(OK.gen(1))
abar

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(19))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(2) + Integer(5))
>>> k = R.residue_field(P, names=('a',)); (a,) = k._first_ngens(1)
>>> f = k.coerce_map_from(R); f
Ring morphism:
  From: Univariate Polynomial Ring in t over Finite Field of size 19
  To:   Residue field in a of Principal ideal (t^2 + 5) of
        Univariate Polynomial Ring in t over Finite Field of size 19
lift(x)[source]#

Returns a lift of x to the Order, returning a “polynomial” in the generator with coefficients between 0 and \(p-1\).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)
sage: P = K.ideal(29).factor()[0][0]
sage: k = K.residue_field(P)
sage: OK = K.maximal_order()
sage: f = k.coerce_map_from(OK)
sage: c = OK(a)
sage: b = k(a)
sage: f.lift(13*b + 5)
13*a + 5
sage: f.lift(12821*b + 918)
3*a + 19

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + t^2 + 7)
sage: k.<a> = P.residue_field(); f = k.coerce_map_from(R)
sage: f.lift(a^2 + 5*a + 1)
t^2 + 5*t + 1
sage: f(f.lift(a^2 + 5*a + 1)) == a^2 + 5*a + 1                             # needs sage.modules
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]
>>> k = K.residue_field(P)
>>> OK = K.maximal_order()
>>> f = k.coerce_map_from(OK)
>>> c = OK(a)
>>> b = k(a)
>>> f.lift(Integer(13)*b + Integer(5))
13*a + 5
>>> f.lift(Integer(12821)*b + Integer(918))
3*a + 19

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(17))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(3) + t**Integer(2) + Integer(7))
>>> k = P.residue_field(names=('a',)); (a,) = k._first_ngens(1); f = k.coerce_map_from(R)
>>> f.lift(a**Integer(2) + Integer(5)*a + Integer(1))
t^2 + 5*t + 1
>>> f(f.lift(a**Integer(2) + Integer(5)*a + Integer(1))) == a**Integer(2) + Integer(5)*a + Integer(1)                             # needs sage.modules
True
section()[source]#

Computes a section of the map, namely a map that lifts elements of the residue field to elements of the ring of integers.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^5 - 5*x + 2)
sage: P = K.ideal(47).factor()[0][0]
sage: k = K.residue_field(P)
sage: f = k.coerce_map_from(K.ring_of_integers())
sage: s = f.section(); s
Lifting map:
  From: Residue field in abar of
        Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
  To:   Maximal Order generated by a in Number Field in a with defining polynomial x^5 - 5*x + 2
sage: s(k.gen())
a
sage: L.<b> = NumberField(x^5 + 17*x + 1)
sage: P = L.factor(53)[0][0]
sage: l = L.residue_field(P)
sage: g = l.coerce_map_from(L.ring_of_integers())
sage: s = g.section(); s
Lifting map:
  From: Residue field in bbar of Fractional ideal (53, b^2 + 23*b + 8)
  To:   Maximal Order generated by b in Number Field in b
        with defining polynomial x^5 + 17*x + 1
sage: s(l.gen()).parent()
Maximal Order generated by b in Number Field in b with defining polynomial x^5 + 17*x + 1

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + t^2 + 7)
sage: k.<a> = P.residue_field()
sage: f = k.coerce_map_from(R)
sage: f.section()
(map internal to coercion system -- copy before use)
Lifting map:
  From: Residue field in a of Principal ideal (t^3 + t^2 + 7) of
        Univariate Polynomial Ring in t over Finite Field of size 17
  To:   Univariate Polynomial Ring in t over Finite Field of size 17

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(5) - Integer(5)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(47)).factor()[Integer(0)][Integer(0)]
>>> k = K.residue_field(P)
>>> f = k.coerce_map_from(K.ring_of_integers())
>>> s = f.section(); s
Lifting map:
  From: Residue field in abar of
        Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
  To:   Maximal Order generated by a in Number Field in a with defining polynomial x^5 - 5*x + 2
>>> s(k.gen())
a
>>> L = NumberField(x**Integer(5) + Integer(17)*x + Integer(1), names=('b',)); (b,) = L._first_ngens(1)
>>> P = L.factor(Integer(53))[Integer(0)][Integer(0)]
>>> l = L.residue_field(P)
>>> g = l.coerce_map_from(L.ring_of_integers())
>>> s = g.section(); s
Lifting map:
  From: Residue field in bbar of Fractional ideal (53, b^2 + 23*b + 8)
  To:   Maximal Order generated by b in Number Field in b
        with defining polynomial x^5 + 17*x + 1
>>> s(l.gen()).parent()
Maximal Order generated by b in Number Field in b with defining polynomial x^5 + 17*x + 1

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(17))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(3) + t**Integer(2) + Integer(7))
>>> k = P.residue_field(names=('a',)); (a,) = k._first_ngens(1)
>>> f = k.coerce_map_from(R)
>>> f.section()
(map internal to coercion system -- copy before use)
Lifting map:
  From: Residue field in a of Principal ideal (t^3 + t^2 + 7) of
        Univariate Polynomial Ring in t over Finite Field of size 17
  To:   Univariate Polynomial Ring in t over Finite Field of size 17
class sage.rings.finite_rings.residue_field.ResidueField_generic(p)[source]#

Bases: Field

The class representing a generic residue field.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: I = QQ[i].factor(2)[0][0]; I
Fractional ideal (I + 1)
sage: k = I.residue_field(); k
Residue field of Fractional ideal (I + 1)
sage: type(k)
<class 'sage.rings.finite_rings.residue_field.ResidueFiniteField_prime_modn_with_category'>

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(29)[]; P = R.ideal(t^2 + 2); k.<a> = ResidueField(P); k
Residue field in a of Principal ideal (t^2 + 2) of
 Univariate Polynomial Ring in t over Finite Field of size 29
sage: type(k)                                                                   # needs sage.libs.linbox
<class 'sage.rings.finite_rings.residue_field_givaro.ResidueFiniteField_givaro_with_category'>

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> I = QQ[i].factor(Integer(2))[Integer(0)][Integer(0)]; I
Fractional ideal (I + 1)
>>> k = I.residue_field(); k
Residue field of Fractional ideal (I + 1)
>>> type(k)
<class 'sage.rings.finite_rings.residue_field.ResidueFiniteField_prime_modn_with_category'>

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(29))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(2) + Integer(2)); k = ResidueField(P, names=('a',)); (a,) = k._first_ngens(1); k
Residue field in a of Principal ideal (t^2 + 2) of
 Univariate Polynomial Ring in t over Finite Field of size 29
>>> type(k)                                                                   # needs sage.libs.linbox
<class 'sage.rings.finite_rings.residue_field_givaro.ResidueFiniteField_givaro_with_category'>
construction()[source]#

Construction of this residue field.

OUTPUT:

An AlgebraicExtensionFunctor and the number field that this residue field has been obtained from.

The residue field is determined by a prime (fractional) ideal in a number field. If this ideal can be coerced into a different number field, then the construction functor applied to this number field will return the corresponding residue field. See Issue #15223.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K.<z> = CyclotomicField(7)
sage: P = K.factor(17)[0][0]
sage: k = K.residue_field(P); k
Residue field in zbar of Fractional ideal (17)
sage: F, R = k.construction()
sage: F
AlgebraicExtensionFunctor
sage: R
Cyclotomic Field of order 7 and degree 6
sage: F(R) is k
True
sage: F(ZZ)
Residue field of Integers modulo 17
sage: F(CyclotomicField(49))
Residue field in zbar of Fractional ideal (17)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(7), names=('z',)); (z,) = K._first_ngens(1)
>>> P = K.factor(Integer(17))[Integer(0)][Integer(0)]
>>> k = K.residue_field(P); k
Residue field in zbar of Fractional ideal (17)
>>> F, R = k.construction()
>>> F
AlgebraicExtensionFunctor
>>> R
Cyclotomic Field of order 7 and degree 6
>>> F(R) is k
True
>>> F(ZZ)
Residue field of Integers modulo 17
>>> F(CyclotomicField(Integer(49)))
Residue field in zbar of Fractional ideal (17)
ideal()[source]#

Return the maximal ideal that this residue field is the quotient by.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + x + 1)
sage: P = K.ideal(29).factor()[0][0]
sage: k = K.residue_field(P)  # indirect doctest
sage: k.ideal() is P
True
sage: p = next_prime(2^40); p
1099511627791
sage: k = K.residue_field(K.prime_above(p))
sage: k.ideal().norm() == p
True

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + t^2 + 7)
sage: k.<a> = R.residue_field(P)
sage: k.ideal()
Principal ideal (t^3 + t^2 + 7) of
 Univariate Polynomial Ring in t over Finite Field of size 17

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]
>>> k = K.residue_field(P)  # indirect doctest
>>> k.ideal() is P
True
>>> p = next_prime(Integer(2)**Integer(40)); p
1099511627791
>>> k = K.residue_field(K.prime_above(p))
>>> k.ideal().norm() == p
True

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(17))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(3) + t**Integer(2) + Integer(7))
>>> k = R.residue_field(P, names=('a',)); (a,) = k._first_ngens(1)
>>> k.ideal()
Principal ideal (t^3 + t^2 + 7) of
 Univariate Polynomial Ring in t over Finite Field of size 17
lift(x)[source]#

Returns a lift of x to the Order, returning a “polynomial” in the generator with coefficients between 0 and \(p-1\).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)
sage: P = K.ideal(29).factor()[0][0]
sage: k = K.residue_field(P)
sage: OK = K.maximal_order()
sage: c = OK(a)
sage: b = k(a)
sage: k.lift(13*b + 5)
13*a + 5
sage: k.lift(12821*b + 918)
3*a + 19

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + t^2 + 7)
sage: k.<a> = P.residue_field()
sage: k.lift(a^2 + 5)
t^2 + 5

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]
>>> k = K.residue_field(P)
>>> OK = K.maximal_order()
>>> c = OK(a)
>>> b = k(a)
>>> k.lift(Integer(13)*b + Integer(5))
13*a + 5
>>> k.lift(Integer(12821)*b + Integer(918))
3*a + 19

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(17))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(3) + t**Integer(2) + Integer(7))
>>> k = P.residue_field(names=('a',)); (a,) = k._first_ngens(1)
>>> k.lift(a**Integer(2) + Integer(5))
t^2 + 5
lift_map()[source]#

Returns the standard map from this residue field up to the ring of integers lifting the canonical projection.

EXAMPLES:

sage: # needs sage.rings.number_field sage.symbolic
sage: I = QQ[3^(1/3)].factor(5)[1][0]; I
Fractional ideal (a - 2)
sage: k = I.residue_field(); k
Residue field of Fractional ideal (a - 2)
sage: f = k.lift_map(); f
Lifting map:
  From: Residue field of Fractional ideal (a - 2)
  To:   Maximal Order generated by a in Number Field in a
        with defining polynomial x^3 - 3 with a = 1.442249570307409?
sage: f.domain()
Residue field of Fractional ideal (a - 2)
sage: f.codomain()
Maximal Order generated by a in Number Field in a
 with defining polynomial x^3 - 3 with a = 1.442249570307409?
sage: f(k.0)
1

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + t^2 + 7)
sage: k.<a> = P.residue_field()
sage: f = k.lift_map(); f
(map internal to coercion system -- copy before use)
Lifting map:
  From: Residue field in a of Principal ideal (t^3 + t^2 + 7) of
        Univariate Polynomial Ring in t over Finite Field of size 17
  To:   Univariate Polynomial Ring in t over Finite Field of size 17
sage: f(a^2 + 5)
t^2 + 5

>>> from sage.all import *
>>> # needs sage.rings.number_field sage.symbolic
>>> I = QQ[Integer(3)**(Integer(1)/Integer(3))].factor(Integer(5))[Integer(1)][Integer(0)]; I
Fractional ideal (a - 2)
>>> k = I.residue_field(); k
Residue field of Fractional ideal (a - 2)
>>> f = k.lift_map(); f
Lifting map:
  From: Residue field of Fractional ideal (a - 2)
  To:   Maximal Order generated by a in Number Field in a
        with defining polynomial x^3 - 3 with a = 1.442249570307409?
>>> f.domain()
Residue field of Fractional ideal (a - 2)
>>> f.codomain()
Maximal Order generated by a in Number Field in a
 with defining polynomial x^3 - 3 with a = 1.442249570307409?
>>> f(k.gen(0))
1

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(17))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(3) + t**Integer(2) + Integer(7))
>>> k = P.residue_field(names=('a',)); (a,) = k._first_ngens(1)
>>> f = k.lift_map(); f
(map internal to coercion system -- copy before use)
Lifting map:
  From: Residue field in a of Principal ideal (t^3 + t^2 + 7) of
        Univariate Polynomial Ring in t over Finite Field of size 17
  To:   Univariate Polynomial Ring in t over Finite Field of size 17
>>> f(a**Integer(2) + Integer(5))
t^2 + 5
reduction_map()[source]#

Return the partially defined reduction map from the number field to this residue class field.

EXAMPLES:

sage: # needs sage.rings.number_field sage.symbolic
sage: I = QQ[2^(1/3)].factor(2)[0][0]; I
Fractional ideal (a)
sage: k = I.residue_field(); k
Residue field of Fractional ideal (a)
sage: pi = k.reduction_map(); pi
Partially defined reduction map:
  From: Number Field in a with defining polynomial x^3 - 2
        with a = 1.259921049894873?
  To:   Residue field of Fractional ideal (a)
sage: pi.domain()
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
sage: pi.codomain()
Residue field of Fractional ideal (a)

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 32)
sage: F = K.factor(2)[0][0].residue_field()
sage: F.reduction_map().domain()
Number Field in a with defining polynomial x^3 + x^2 - 2*x + 32
sage: K.<a> = NumberField(x^3 + 128)
sage: F = K.factor(2)[0][0].residue_field()
sage: F.reduction_map().codomain()
Residue field of Fractional ideal (1/4*a)

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + t^2 + 7)
sage: k.<a> = P.residue_field(); f = k.reduction_map(); f
Partially defined reduction map:
  From: Fraction Field of Univariate Polynomial Ring in t
        over Finite Field of size 17
  To:   Residue field in a of Principal ideal (t^3 + t^2 + 7) of
        Univariate Polynomial Ring in t over Finite Field of size 17
sage: f(1/t)
12*a^2 + 12*a

>>> from sage.all import *
>>> # needs sage.rings.number_field sage.symbolic
>>> I = QQ[Integer(2)**(Integer(1)/Integer(3))].factor(Integer(2))[Integer(0)][Integer(0)]; I
Fractional ideal (a)
>>> k = I.residue_field(); k
Residue field of Fractional ideal (a)
>>> pi = k.reduction_map(); pi
Partially defined reduction map:
  From: Number Field in a with defining polynomial x^3 - 2
        with a = 1.259921049894873?
  To:   Residue field of Fractional ideal (a)
>>> pi.domain()
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
>>> pi.codomain()
Residue field of Fractional ideal (a)

>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(32), names=('a',)); (a,) = K._first_ngens(1)
>>> F = K.factor(Integer(2))[Integer(0)][Integer(0)].residue_field()
>>> F.reduction_map().domain()
Number Field in a with defining polynomial x^3 + x^2 - 2*x + 32
>>> K = NumberField(x**Integer(3) + Integer(128), names=('a',)); (a,) = K._first_ngens(1)
>>> F = K.factor(Integer(2))[Integer(0)][Integer(0)].residue_field()
>>> F.reduction_map().codomain()
Residue field of Fractional ideal (1/4*a)

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(17))['t']; (t,) = R._first_ngens(1); P = R.ideal(t**Integer(3) + t**Integer(2) + Integer(7))
>>> k = P.residue_field(names=('a',)); (a,) = k._first_ngens(1); f = k.reduction_map(); f
Partially defined reduction map:
  From: Fraction Field of Univariate Polynomial Ring in t
        over Finite Field of size 17
  To:   Residue field in a of Principal ideal (t^3 + t^2 + 7) of
        Univariate Polynomial Ring in t over Finite Field of size 17
>>> f(Integer(1)/t)
12*a^2 + 12*a
class sage.rings.finite_rings.residue_field.ResidueFiniteField_prime_modn(p, name, intp, to_vs, to_order, PB)[source]#

Bases: ResidueField_generic, FiniteField_prime_modn

The class representing residue fields of number fields that have prime order.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3 - 7)
sage: P = K.ideal(29).factor()[1][0]
sage: k = ResidueField(P); k
Residue field of Fractional ideal (-a^2 - 2*a - 2)
sage: k.order()
29
sage: OK = K.maximal_order()
sage: c = OK(a)
sage: b = k(a)
sage: k.coerce_map_from(OK)(c)
16
sage: k(4)
4
sage: k(c + 5)
21
sage: b + c
3

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(7)[]; P = R.ideal(2*t + 3)
sage: k = P.residue_field(); k
Residue field of Principal ideal (t + 5) of
 Univariate Polynomial Ring in t over Finite Field of size 7
sage: k(t^2)
4
sage: k.order()
7

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(29)).factor()[Integer(1)][Integer(0)]
>>> k = ResidueField(P); k
Residue field of Fractional ideal (-a^2 - 2*a - 2)
>>> k.order()
29
>>> OK = K.maximal_order()
>>> c = OK(a)
>>> b = k(a)
>>> k.coerce_map_from(OK)(c)
16
>>> k(Integer(4))
4
>>> k(c + Integer(5))
21
>>> b + c
3

>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(7))['t']; (t,) = R._first_ngens(1); P = R.ideal(Integer(2)*t + Integer(3))
>>> k = P.residue_field(); k
Residue field of Principal ideal (t + 5) of
 Univariate Polynomial Ring in t over Finite Field of size 7
>>> k(t**Integer(2))
4
>>> k.order()
7