Orders of function fields: basis

class sage.rings.function_field.order_basis.FunctionFieldOrderInfinite_basis(basis, check=True)[source]

Bases: FunctionFieldOrderInfinite

Order given by a basis over the infinite maximal order of the base field.

INPUT:

  • basis – elements of the function field

  • check – boolean (default: True); if True, check the basis generates an order

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)                                  # needs sage.rings.function_field
sage: O = L.equation_order_infinite(); O                                        # needs sage.rings.function_field
Infinite order in Function field in y defined by y^4 + x*y + 4*x + 1

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> O = L.equation_order_infinite(); O                                        # needs sage.rings.function_field
Infinite order in Function field in y defined by y^4 + x*y + 4*x + 1

The basis only defines an order if the module it generates is closed under multiplication and contains the identity element (only checked when check is True):

sage: O = L.order_infinite_with_basis([1, y, 1/x^2*y^2, y^3]); O                # needs sage.rings.function_field
Traceback (most recent call last):
...
ValueError: the module generated by basis (1, y, 1/x^2*y^2, y^3)
must be closed under multiplication

>>> from sage.all import *
>>> O = L.order_infinite_with_basis([Integer(1), y, Integer(1)/x**Integer(2)*y**Integer(2), y**Integer(3)]); O                # needs sage.rings.function_field
Traceback (most recent call last):
...
ValueError: the module generated by basis (1, y, 1/x^2*y^2, y^3)
must be closed under multiplication

The basis also has to be linearly independent and of the same rank as the degree of the function field of its elements (only checked when check is True):

sage: O = L.order_infinite_with_basis([1, y, 1/x^2*y^2, 1 + y]); O              # needs sage.rings.function_field
Traceback (most recent call last):
...
ValueError: The given basis vectors must be linearly independent.

>>> from sage.all import *
>>> O = L.order_infinite_with_basis([Integer(1), y, Integer(1)/x**Integer(2)*y**Integer(2), Integer(1) + y]); O              # needs sage.rings.function_field
Traceback (most recent call last):
...
ValueError: The given basis vectors must be linearly independent.

Note that 1 does not need to be an element of the basis, as long as it is in the module spanned by it:

sage: # needs sage.rings.function_field
sage: O = L.order_infinite_with_basis([1 + 1/x*y, 1/x*y, 1/x^2*y^2, 1/x^3*y^3]); O
Infinite order in Function field in y defined by y^4 + x*y + 4*x + 1
sage: O.basis()
(1/x*y + 1, 1/x*y, 1/x^2*y^2, 1/x^3*y^3)

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> O = L.order_infinite_with_basis([Integer(1) + Integer(1)/x*y, Integer(1)/x*y, Integer(1)/x**Integer(2)*y**Integer(2), Integer(1)/x**Integer(3)*y**Integer(3)]); O
Infinite order in Function field in y defined by y^4 + x*y + 4*x + 1
>>> O.basis()
(1/x*y + 1, 1/x*y, 1/x^2*y^2, 1/x^3*y^3)
basis()[source]

Return a basis of this order over the maximal order of the base field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)                              # needs sage.rings.function_field
sage: O = L.equation_order()                                                # needs sage.rings.function_field
sage: O.basis()                                                             # needs sage.rings.function_field
(1, y, y^2, y^3)

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> O = L.equation_order()                                                # needs sage.rings.function_field
>>> O.basis()                                                             # needs sage.rings.function_field
(1, y, y^2, y^3)
free_module()[source]

Return the free module formed by the basis over the maximal order of the base field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)                              # needs sage.rings.function_field
sage: O = L.equation_order()                                                # needs sage.rings.function_field
sage: O.free_module()                                                       # needs sage.rings.function_field
Free module of degree 4 and rank 4 over Maximal order of Rational
function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> O = L.equation_order()                                                # needs sage.rings.function_field
>>> O.free_module()                                                       # needs sage.rings.function_field
Free module of degree 4 and rank 4 over Maximal order of Rational
function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
ideal(*gens)[source]

Return the fractional ideal generated by the elements in gens.

INPUT:

  • gens – list of generators or an ideal in a ring which coerces to this order

EXAMPLES:

sage: K.<y> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: O.ideal(y)
Ideal (y) of Maximal order of Rational function field in y over Rational Field
sage: O.ideal([y,1/y]) == O.ideal(y,1/y) # multiple generators may be given as a list
True

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('y',)); (y,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> O.ideal(y)
Ideal (y) of Maximal order of Rational function field in y over Rational Field
>>> O.ideal([y,Integer(1)/y]) == O.ideal(y,Integer(1)/y) # multiple generators may be given as a list
True

A fractional ideal of a nontrivial extension:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: O = K.maximal_order_infinite()
sage: I = O.ideal(x^2 - 4)
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                    # needs sage.rings.function_field
sage: S = L.order_infinite_with_basis([1, 1/x^2*y])                         # needs sage.rings.function_field

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> O = K.maximal_order_infinite()
>>> I = O.ideal(x**Integer(2) - Integer(4))
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> S = L.order_infinite_with_basis([Integer(1), Integer(1)/x**Integer(2)*y])                         # needs sage.rings.function_field
ideal_with_gens_over_base(gens)[source]

Return the fractional ideal with basis gens over the maximal order of the base field.

It is not checked that gens really generates an ideal.

INPUT:

  • gens – list of elements that are a basis for the ideal over the maximal order of the base field

EXAMPLES:

We construct an ideal in a rational function field:

sage: K.<y> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal([y]); I
Ideal (y) of Maximal order of Rational function field in y over Rational Field
sage: I*I
Ideal (y^2) of Maximal order of Rational function field in y over Rational Field

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('y',)); (y,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> I = O.ideal([y]); I
Ideal (y) of Maximal order of Rational function field in y over Rational Field
>>> I*I
Ideal (y^2) of Maximal order of Rational function field in y over Rational Field

We construct some ideals in a nontrivial function field:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order(); O
Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: I = O.ideal_with_gens_over_base([1, y]);  I
Ideal (1) of Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: I.module()
Free module of degree 2 and rank 2 over
 Maximal order of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0]
[0 1]

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order(); O
Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> I = O.ideal_with_gens_over_base([Integer(1), y]);  I
Ideal (1) of Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> I.module()
Free module of degree 2 and rank 2 over
 Maximal order of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0]
[0 1]

There is no check if the resulting object is really an ideal:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal_with_gens_over_base([y]); I
Ideal (y) of Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: y in I
True
sage: y^2 in I
False

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order()
>>> I = O.ideal_with_gens_over_base([y]); I
Ideal (y) of Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> y in I
True
>>> y**Integer(2) in I
False
polynomial()[source]

Return the defining polynomial of the function field of which this is an order.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)                              # needs sage.rings.function_field
sage: O = L.equation_order()                                                # needs sage.rings.function_field
sage: O.polynomial()                                                        # needs sage.rings.function_field
y^4 + x*y + 4*x + 1

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> O = L.equation_order()                                                # needs sage.rings.function_field
>>> O.polynomial()                                                        # needs sage.rings.function_field
y^4 + x*y + 4*x + 1
class sage.rings.function_field.order_basis.FunctionFieldOrder_basis(basis, check=True)[source]

Bases: FunctionFieldOrder

Order given by a basis over the maximal order of the base field.

INPUT:

  • basis – list of elements of the function field

  • check – boolean (default: True); if True, check whether the module that basis generates forms an order

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)                                  # needs sage.rings.function_field
sage: O = L.equation_order(); O                                                 # needs sage.rings.function_field
Order in Function field in y defined by y^4 + x*y + 4*x + 1

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> O = L.equation_order(); O                                                 # needs sage.rings.function_field
Order in Function field in y defined by y^4 + x*y + 4*x + 1

The basis only defines an order if the module it generates is closed under multiplication and contains the identity element:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x))                            # needs sage.rings.function_field
sage: y.is_integral()                                                           # needs sage.rings.function_field
False
sage: L.order(y)                                                                # needs sage.rings.function_field
Traceback (most recent call last):
...
ValueError: the module generated by basis (1, y, y^2, y^3, y^4)
must be closed under multiplication

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1)
>>> R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> y.is_integral()                                                           # needs sage.rings.function_field
False
>>> L.order(y)                                                                # needs sage.rings.function_field
Traceback (most recent call last):
...
ValueError: the module generated by basis (1, y, y^2, y^3, y^4)
must be closed under multiplication

The basis also has to be linearly independent and of the same rank as the degree of the function field of its elements (only checked when check is True):

sage: # needs sage.rings.function_field
sage: L.order(L(x))
Traceback (most recent call last):
...
ValueError: basis (1, x, x^2, x^3, x^4) is not linearly independent
sage: from sage.rings.function_field.order_basis import FunctionFieldOrder_basis
sage: FunctionFieldOrder_basis((y,y,y^3,y^4,y^5))
Traceback (most recent call last):
...
ValueError: basis (y, y, y^3, y^4, 2*x*y + (x^4 + 1)/x) is not linearly independent

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> L.order(L(x))
Traceback (most recent call last):
...
ValueError: basis (1, x, x^2, x^3, x^4) is not linearly independent
>>> from sage.rings.function_field.order_basis import FunctionFieldOrder_basis
>>> FunctionFieldOrder_basis((y,y,y**Integer(3),y**Integer(4),y**Integer(5)))
Traceback (most recent call last):
...
ValueError: basis (y, y, y^3, y^4, 2*x*y + (x^4 + 1)/x) is not linearly independent
basis()[source]

Return a basis of the order over the maximal order of the base field.

EXAMPLES:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.equation_order()
sage: O.basis()
(1, y, y^2, y^3)

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order()
>>> O.basis()
(1, y, y^2, y^3)
coordinate_vector(e)[source]

Return the coordinates of e with respect to the basis of the order.

INPUT:

  • e – element of the order or the function field

EXAMPLES:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.equation_order()
sage: f = (x + y)^3
sage: O.coordinate_vector(f)
(x^3, 3*x^2, 3*x, 1)

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order()
>>> f = (x + y)**Integer(3)
>>> O.coordinate_vector(f)
(x^3, 3*x^2, 3*x, 1)
free_module()[source]

Return the free module formed by the basis over the maximal order of the base function field.

EXAMPLES:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.equation_order()
sage: O.free_module()
Free module of degree 4 and rank 4 over Maximal order of Rational
function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order()
>>> O.free_module()
Free module of degree 4 and rank 4 over Maximal order of Rational
function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
ideal(*gens)[source]

Return the fractional ideal generated by the elements in gens.

INPUT:

  • gens – list of generators or an ideal in a ring which coerces to this order

EXAMPLES:

sage: K.<y> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: O.ideal(y)
Ideal (y) of Maximal order of Rational function field in y over Rational Field
sage: O.ideal([y,1/y]) == O.ideal(y,1/y) # multiple generators may be given as a list
True

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('y',)); (y,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> O.ideal(y)
Ideal (y) of Maximal order of Rational function field in y over Rational Field
>>> O.ideal([y,Integer(1)/y]) == O.ideal(y,Integer(1)/y) # multiple generators may be given as a list
True

A fractional ideal of a nontrivial extension:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: O = K.maximal_order()
sage: I = O.ideal(x^2 - 4)

sage: # needs sage.rings.function_field
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: S = L.equation_order()
sage: S.ideal(1/y)
Ideal (1, (6/(x^3 + 1))*y) of
 Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: I2 = S.ideal(x^2 - 4); I2
Ideal (x^2 + 3) of Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: I2 == S.ideal(I)
True

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> O = K.maximal_order()
>>> I = O.ideal(x**Integer(2) - Integer(4))

>>> # needs sage.rings.function_field
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> S = L.equation_order()
>>> S.ideal(Integer(1)/y)
Ideal (1, (6/(x^3 + 1))*y) of
 Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> I2 = S.ideal(x**Integer(2) - Integer(4)); I2
Ideal (x^2 + 3) of Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> I2 == S.ideal(I)
True
ideal_with_gens_over_base(gens)[source]

Return the fractional ideal with basis gens over the maximal order of the base field.

It is not checked that the gens really generates an ideal.

INPUT:

  • gens – list of elements of the function field

EXAMPLES:

We construct an ideal in a rational function field:

sage: K.<y> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal([y]); I
Ideal (y) of Maximal order of Rational function field in y over Rational Field
sage: I * I
Ideal (y^2) of Maximal order of Rational function field in y over Rational Field

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('y',)); (y,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> I = O.ideal([y]); I
Ideal (y) of Maximal order of Rational function field in y over Rational Field
>>> I * I
Ideal (y^2) of Maximal order of Rational function field in y over Rational Field

We construct some ideals in a nontrivial function field:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order(); O
Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: I = O.ideal_with_gens_over_base([1, y]);  I
Ideal (1) of Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: I.module()
Free module of degree 2 and rank 2 over
 Maximal order of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0]
[0 1]

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order(); O
Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> I = O.ideal_with_gens_over_base([Integer(1), y]);  I
Ideal (1) of Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> I.module()
Free module of degree 2 and rank 2 over
 Maximal order of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[1 0]
[0 1]

There is no check if the resulting object is really an ideal:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal_with_gens_over_base([y]); I
Ideal (y) of Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: y in I
True
sage: y^2 in I
False

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order()
>>> I = O.ideal_with_gens_over_base([y]); I
Ideal (y) of Order in Function field in y defined by y^2 + 6*x^3 + 6
>>> y in I
True
>>> y**Integer(2) in I
False
polynomial()[source]

Return the defining polynomial of the function field of which this is an order.

EXAMPLES:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.equation_order()
sage: O.polynomial()
y^4 + x*y + 4*x + 1

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(4) + x*y + Integer(4)*x + Integer(1), names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.equation_order()
>>> O.polynomial()
y^4 + x*y + 4*x + 1