Orders of function fields: extension

class sage.rings.function_field.order_polymod.FunctionFieldMaximalOrderInfinite_polymod(field, category=None)

Bases: FunctionFieldMaximalOrderInfinite

Maximal infinite orders of function fields.

INPUT:

• field – function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)                   # needs sage.rings.finite_rings
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)                                # needs sage.rings.finite_rings
sage: F.maximal_order_infinite()                                                # needs sage.rings.finite_rings
Maximal infinite order of Function field in y defined by y^3 + x^6 + x^4 + x^2

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                    # needs sage.rings.finite_rings
sage: L.maximal_order_infinite()                                                # needs sage.rings.finite_rings
Maximal infinite order of Function field in y defined by y^2 + y + (x^2 + 1)/x

basis()

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: L.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.basis()
(1, 1/x^2*y, (1/(x^4 + x^3 + x^2))*y^2)

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.basis()
(1, 1/x*y)

coordinate_vector(e)

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

INPUT:

• e – element of the function field

The returned coordinates are in the base maximal infinite order if and only if the element is in the order.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: f = 1/y^2
sage: f in Oinf
True
sage: Oinf.coordinate_vector(f)
((x^3 + x^2 + x)/(x^4 + 1), x^3/(x^4 + 1))

decomposition()

Return prime ideal decomposition of \(pO_\infty\) where \(p\) is the unique prime ideal of the maximal infinite order of the rational function field.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = F.maximal_order_infinite()
sage: Oinf.decomposition()
[(Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1) of Maximal infinite order
 of Function field in y defined by y^3 + x^6 + x^4 + x^2, 1, 1),
 (Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1/x^2*y + 1) of Maximal infinite order
 of Function field in y defined by y^3 + x^6 + x^4 + x^2, 2, 1)]

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.decomposition()
[(Ideal (1/x*y) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x, 1, 2)]

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = F.maximal_order_infinite()
sage: Oinf.decomposition()
[(Ideal (1/x^2*y - 1) of Maximal infinite order
 of Function field in y defined by y^3 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 - x^2, 1, 1),
 (Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1/x^2*y + 1) of Maximal infinite order
 of Function field in y defined by y^3 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 - x^2, 2, 1)]

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.decomposition()
[(Ideal (1/x*y) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x, 1, 2)]

different()

Return the different ideal of the maximal infinite order.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.different()
Ideal (1/x) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x

gen(n=0)

Return the n-th generator of the order.

The basis elements of the order are generators.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: L.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.gen()
1
sage: Oinf.gen(1)
1/x^2*y
sage: Oinf.gen(2)
(1/(x^4 + x^3 + x^2))*y^2
sage: Oinf.gen(3)
Traceback (most recent call last):
...
IndexError: there are only 3 generators

ideal(*gens)

Return the ideal generated by gens.

INPUT:

• gens – tuple of elements of the function field

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x, y); I
Ideal (y) of Maximal infinite order of Function field
in y defined by y^3 + x^6 + x^4 + x^2

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(x, y); I
Ideal (x) of Maximal infinite order of Function field
in y defined by y^2 + y + (x^2 + 1)/x

ideal_with_gens_over_base(gens)

Return the ideal generated by gens as a module.

INPUT:

• gens – tuple of elements of the function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); R.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = F.maximal_order_infinite()
sage: Oinf.ideal_with_gens_over_base((x^2, y, (1/(x^2 + x + 1))*y^2))
Ideal (y) of Maximal infinite order of Function field in y
defined by y^3 + x^6 + x^4 + x^2

ngens()

Return the number of generators of the order.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: L.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = L.maximal_order_infinite()
sage: Oinf.ngens()
3

class sage.rings.function_field.order_polymod.FunctionFieldMaximalOrder_global(field)

Bases: FunctionFieldMaximalOrder_polymod

Maximal orders of global function fields.

INPUT:

• field – function field to which this maximal order belongs

EXAMPLES:

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

decomposition(ideal)

Return the decomposition of the prime ideal.

INPUT:

• ideal – prime ideal of the base maximal order

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); R.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: o = K.maximal_order()
sage: O = F.maximal_order()
sage: p = o.ideal(x + 1)
sage: O.decomposition(p)
[(Ideal (x + 1, y + 1) of Maximal order
 of Function field in y defined by y^3 + x^6 + x^4 + x^2, 1, 1),
 (Ideal (x + 1, (1/(x^3 + x^2 + x))*y^2 + y + 1) of Maximal order
 of Function field in y defined by y^3 + x^6 + x^4 + x^2, 2, 1)]

p_radical(prime)

Return the prime-radical of the maximal order.

INPUT:

• prime – prime ideal of the maximal order of the base rational function field

The algorithm is outlined in Section 6.1.3 of [Coh1993].

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2 * (x^2 + x + 1)^2)
sage: o = K.maximal_order()
sage: O = F.maximal_order()
sage: p = o.ideal(x + 1)
sage: O.p_radical(p)
Ideal (x + 1) of Maximal order of Function field in y
defined by y^3 + x^6 + x^4 + x^2

class sage.rings.function_field.order_polymod.FunctionFieldMaximalOrder_polymod(field, ideal_class=<class 'sage.rings.function_field.ideal_polymod.FunctionFieldIdeal_polymod'>)

Bases: FunctionFieldMaximalOrder

Maximal orders of extensions of function fields.

basis()

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
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)

sage: K.<x> = FunctionField(QQ)
sage: R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^4 + x^12*t^2 + x^18*t + x^21 + x^18)
sage: O = F.maximal_order()
sage: O.basis()
(1, 1/x^4*y, 1/x^9*y^2, 1/x^13*y^3)

codifferent()

Return the codifferent ideal of the function field.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.maximal_order()
sage: O.codifferent()
Ideal (1, (1/(x^4 + 4*x^3 + 3*x^2 + 6*x + 4))*y^3
+ ((5*x^3 + 6*x^2 + x + 6)/(x^4 + 4*x^3 + 3*x^2 + 6*x + 4))*y^2
+ ((x^3 + 2*x^2 + 2*x + 2)/(x^4 + 4*x^3 + 3*x^2 + 6*x + 4))*y
+ 6*x/(x^4 + 4*x^3 + 3*x^2 + 6*x + 4)) of Maximal order of Function field
in y defined by y^4 + x*y + 4*x + 1

coordinate_vector(e)

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.maximal_order()
sage: O.coordinate_vector(y)
(0, 1, 0, 0)
sage: O.coordinate_vector(x*y)
(0, x, 0, 0)

sage: K.<x> = FunctionField(QQ); 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)

decomposition(ideal)

Return the decomposition of the prime ideal.

INPUT:

• ideal – prime ideal of the base maximal order

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); R.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: o = K.maximal_order()
sage: O = F.maximal_order()
sage: p = o.ideal(x + 1)
sage: O.decomposition(p)
[(Ideal (x + 1, y + 1) of Maximal order
 of Function field in y defined by y^3 + x^6 + x^4 + x^2, 1, 1),
 (Ideal (x + 1, (1/(x^3 + x^2 + x))*y^2 + y + 1) of Maximal order
 of Function field in y defined by y^3 + x^6 + x^4 + x^2, 2, 1)]

ALGORITHM:

In principle, we’re trying to compute a primary decomposition of the extension of ideal in self (an order, and therefore a ring). However, while we have primary decomposition methods for polynomial rings, we lack any such method for an order. Therefore, we construct self mod ideal as a finite-dimensional algebra, a construct for which we do support primary decomposition.

See github issue #28094 and https://github.com/sagemath/sage/files/10659303/decomposition.pdf.gz

Todo

Use Kummer’s theorem to shortcut this code if possible, like as done in FunctionFieldMaximalOrder_global.decomposition()

different()

Return the different ideal of the function field.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.maximal_order()
sage: O.different()
Ideal (y^3 + 2*x)
of Maximal order of Function field in y defined by y^4 + x*y + 4*x + 1

free_module()

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^4 + x*y + 4*x + 1)
sage: O = L.maximal_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
User basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

gen(n=0)

Return the n-th generator of the order.

The basis elements of the order are generators.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: L.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: O = L.maximal_order()
sage: O.gen()
1
sage: O.gen(1)
y
sage: O.gen(2)
(1/(x^3 + x^2 + x))*y^2
sage: O.gen(3)
Traceback (most recent call last):
...
IndexError: there are only 3 generators

ideal(*gens, **kwargs)

Return the fractional ideal generated by the elements in gens.

INPUT:

• gens – list of generators

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: O = K.maximal_order()
sage: I = O.ideal(x^2 - 4)
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: S = L.maximal_order()
sage: S.ideal(1/y)
Ideal ((1/(x^3 + 1))*y) of Maximal order of 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 Maximal order of Function field in y defined by y^2 + 6*x^3 + 6
sage: I2 == S.ideal(I)
True

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: O = K.maximal_order()
sage: I = O.ideal(x^2 - 4)
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: S = L.maximal_order()
sage: S.ideal(1/y)
Ideal ((1/(x^3 + 1))*y) of
 Maximal order of Function field in y defined by y^2 - x^3 - 1
sage: I2 = S.ideal(x^2-4); I2
Ideal (x^2 - 4) of Maximal order of Function field in y defined by y^2 - x^3 - 1
sage: I2 == S.ideal(I)
True

ideal_with_gens_over_base(gens)

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

INPUT:

• gens – list of elements that generates the ideal over the maximal order of the base field

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order(); O
Maximal order of 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 Maximal order of 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]

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

sage: # needs sage.rings.finite_rings
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

ngens()

Return the number of generators of the order.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: L.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: Oinf = L.maximal_order()
sage: Oinf.ngens()
3

polynomial()

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
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

sage: K.<x> = FunctionField(QQ); 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