Places of function fields: extension

class sage.rings.function_field.place_polymod.FunctionFieldPlace_polymod(parent, prime)[source]

Bases: FunctionFieldPlace

Places of extensions of function fields.

degree()[source]

Return the degree of the place.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: OK = K.maximal_order()
sage: OL = L.maximal_order()
sage: p = OK.ideal(x^2 + x + 1)
sage: dec = OL.decomposition(p)
sage: q = dec[0][0].place()
sage: q.degree()
2

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> OK = K.maximal_order()
>>> OL = L.maximal_order()
>>> p = OK.ideal(x**Integer(2) + x + Integer(1))
>>> dec = OL.decomposition(p)
>>> q = dec[Integer(0)][Integer(0)].place()
>>> q.degree()
2

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
OK = K.maximal_order()
OL = L.maximal_order()
p = OK.ideal(x^2 + x + 1)
dec = OL.decomposition(p)
q = dec[0][0].place()
q.degree()
gaps()[source]

Return the gap sequence for the place.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: O = L.maximal_order()
sage: p = O.ideal(x,y).place()
sage: p.gaps() # a Weierstrass place
[1, 2, 4]

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                             # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x^3 * Y + x)                                # needs sage.rings.finite_rings
sage: [p.gaps() for p in L.places()]                                        # needs sage.rings.finite_rings
[[1, 2, 4], [1, 2, 4], [1, 2, 4]]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> O = L.maximal_order()
>>> p = O.ideal(x,y).place()
>>> p.gaps() # a Weierstrass place
[1, 2, 4]

>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings
>>> L = K.extension(Y**Integer(3) + x**Integer(3) * Y + x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings
>>> [p.gaps() for p in L.places()]                                        # needs sage.rings.finite_rings
[[1, 2, 4], [1, 2, 4], [1, 2, 4]]

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(4)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
O = L.maximal_order()
p = O.ideal(x,y).place()
p.gaps() # a Weierstrass place
K.<x> = FunctionField(GF(2)); _.<Y> = K[]                             # needs sage.rings.finite_rings
L.<y> = K.extension(Y^3 + x^3 * Y + x)                                # needs sage.rings.finite_rings
[p.gaps() for p in L.places()]                                        # needs sage.rings.finite_rings
is_infinite_place()[source]

Return True if the place is above the unique infinite place of the underlying rational function field.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: pls = L.places()
sage: [p.is_infinite_place() for p in pls]
[True, True, False]
sage: [p.place_below() for p in pls]
[Place (1/x), Place (1/x), Place (x)]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> pls = L.places()
>>> [p.is_infinite_place() for p in pls]
[True, True, False]
>>> [p.place_below() for p in pls]
[Place (1/x), Place (1/x), Place (x)]

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
pls = L.places()
[p.is_infinite_place() for p in pls]
[p.place_below() for p in pls]
local_uniformizer()[source]

Return an element of the function field that has a simple zero at the place.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: pls = L.places()
sage: [p.local_uniformizer().valuation(p) for p in pls]
[1, 1, 1, 1, 1]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> pls = L.places()
>>> [p.local_uniformizer().valuation(p) for p in pls]
[1, 1, 1, 1, 1]

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(4)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
pls = L.places()
[p.local_uniformizer().valuation(p) for p in pls]
place_below()[source]

Return the place lying below the place.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: OK = K.maximal_order()
sage: OL = L.maximal_order()
sage: p = OK.ideal(x^2 + x + 1)
sage: dec = OL.decomposition(p)
sage: q = dec[0][0].place()
sage: q.place_below()
Place (x^2 + x + 1)

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> OK = K.maximal_order()
>>> OL = L.maximal_order()
>>> p = OK.ideal(x**Integer(2) + x + Integer(1))
>>> dec = OL.decomposition(p)
>>> q = dec[Integer(0)][Integer(0)].place()
>>> q.place_below()
Place (x^2 + x + 1)

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
OK = K.maximal_order()
OL = L.maximal_order()
p = OK.ideal(x^2 + x + 1)
dec = OL.decomposition(p)
q = dec[0][0].place()
q.place_below()
relative_degree()[source]

Return the relative degree of the place.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: OK = K.maximal_order()
sage: OL = L.maximal_order()
sage: p = OK.ideal(x^2 + x + 1)
sage: dec = OL.decomposition(p)
sage: q = dec[0][0].place()
sage: q.relative_degree()
1

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> OK = K.maximal_order()
>>> OL = L.maximal_order()
>>> p = OK.ideal(x**Integer(2) + x + Integer(1))
>>> dec = OL.decomposition(p)
>>> q = dec[Integer(0)][Integer(0)].place()
>>> q.relative_degree()
1

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
OK = K.maximal_order()
OL = L.maximal_order()
p = OK.ideal(x^2 + x + 1)
dec = OL.decomposition(p)
q = dec[0][0].place()
q.relative_degree()
residue_field(name=None)[source]

Return the residue field of the place.

INPUT:

  • name – string; name of the generator of the residue field

OUTPUT:

  • a field isomorphic to the residue field

  • a ring homomorphism from the valuation ring to the field

  • a ring homomorphism from the field to the valuation ring

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: p = L.places_finite()[0]
sage: k, fr_k, to_k = p.residue_field()
sage: k
Finite Field of size 2
sage: fr_k
Ring morphism:
  From: Finite Field of size 2
  To:   Valuation ring at Place (x, x*y)
sage: to_k
Ring morphism:
  From: Valuation ring at Place (x, x*y)
  To:   Finite Field of size 2
sage: to_k(y)
Traceback (most recent call last):
...
TypeError: y fails to convert into the map's domain
Valuation ring at Place (x, x*y)...
sage: to_k(1/y)
0
sage: to_k(y/(1+y))
1

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> p = L.places_finite()[Integer(0)]
>>> k, fr_k, to_k = p.residue_field()
>>> k
Finite Field of size 2
>>> fr_k
Ring morphism:
  From: Finite Field of size 2
  To:   Valuation ring at Place (x, x*y)
>>> to_k
Ring morphism:
  From: Valuation ring at Place (x, x*y)
  To:   Finite Field of size 2
>>> to_k(y)
Traceback (most recent call last):
...
TypeError: y fails to convert into the map's domain
Valuation ring at Place (x, x*y)...
>>> to_k(Integer(1)/y)
0
>>> to_k(y/(Integer(1)+y))
1

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^2 + Y + x + 1/x)
p = L.places_finite()[0]
k, fr_k, to_k = p.residue_field()
k
fr_k
to_k
to_k(y)
to_k(1/y)
to_k(y/(1+y))
valuation_ring()[source]

Return the valuation ring at the place.

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: p = L.places_finite()[0]
sage: p.valuation_ring()
Valuation ring at Place (x, x*y)

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> p = L.places_finite()[Integer(0)]
>>> p.valuation_ring()
Valuation ring at Place (x, x*y)

# needs sage.rings.finite_rings
K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^2 + Y + x + 1/x)
p = L.places_finite()[0]
p.valuation_ring()