Algebraic closures of finite fields¶

Let \(\Bold{F}\) be a finite field, and let \(\overline{\Bold{F}}\) be an algebraic closure of \(\Bold{F}\); this is unique up to (non-canonical) isomorphism. For every \(n\ge 1\), there is a unique subfield \(\Bold{F}_n\) of \(\overline{\Bold{F}}\) such that \(\Bold{F}\subset\Bold{F}_n\) and \([\Bold{F}_n:\Bold{F}]=n\).

In Sage, algebraic closures of finite fields are implemented using compatible systems of finite fields. The resulting Sage object keeps track of a finite lattice of the subfields \(\Bold{F}_n\) and the embeddings between them. This lattice is extended as necessary.

The Sage class corresponding to \(\overline{\Bold{F}}\) can be constructed from the finite field \(\Bold{F}\) by using the algebraic_closure() method.

The Sage class for elements of \(\overline{\Bold{F}}\) is AlgebraicClosureFiniteFieldElement. Such an element is represented as an element of one of the \(\Bold{F}_n\). This means that each element \(x\in\Bold{F}\) has infinitely many different representations, one for each \(n\) such that \(x\) is in \(\Bold{F}_n\).

Note

Only prime finite fields are currently accepted as base fields for algebraic closures. To obtain an algebraic closure of a non-prime finite field \(\Bold{F}\), take an algebraic closure of the prime field of \(\Bold{F}\) and embed \(\Bold{F}\) into this.

Algebraic closures of finite fields are currently implemented using (pseudo-)Conway polynomials; see AlgebraicClosureFiniteField_pseudo_conway and the module conway_polynomials. Other implementations may be added by creating appropriate subclasses of AlgebraicClosureFiniteField_generic.

In the current implementation, algebraic closures do not satisfy the unique parent condition. Moreover, there is no coercion map between different algebraic closures of the same finite field. There is a conceptual reason for this, namely that the definition of pseudo-Conway polynomials only determines an algebraic closure up to non-unique isomorphism. This means in particular that different algebraic closures, and their respective elements, never compare equal.

AUTHORS:

Peter Bruin (August 2013): initial version
Vincent Delecroix (November 2013): additional methods

sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField(base_ring, name, category=None, implementation=None, **kwds)[source]¶

Construct an algebraic closure of a finite field.

The recommended way to use this functionality is by calling the algebraic_closure() method of the finite field.

Note

Algebraic closures of finite fields in Sage do not have the unique representation property, because they are not determined up to unique isomorphism by their defining data.

EXAMPLES:

Sage

sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
sage: F = GF(2).algebraic_closure()
sage: F1 = AlgebraicClosureFiniteField(GF(2), 'z')
sage: F1 is F
False

Python

>>> from sage.all import *
>>> from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
>>> F = GF(Integer(2)).algebraic_closure()
>>> F1 = AlgebraicClosureFiniteField(GF(Integer(2)), 'z')
>>> F1 is F
False

Sage Live

from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
F = GF(2).algebraic_closure()
F1 = AlgebraicClosureFiniteField(GF(2), 'z')
F1 is F

In the pseudo-Conway implementation, non-identical instances never compare equal:

Sage

sage: F1 == F
False
sage: loads(dumps(F)) == F
False

Python

>>> from sage.all import *
>>> F1 == F
False
>>> loads(dumps(F)) == F
False

Sage Live

F1 == F
loads(dumps(F)) == F

This is to ensure that the result of comparing two instances cannot change with time.

class sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteFieldElement(parent, value)[source]¶

Bases: FieldElement

Element of an algebraic closure of a finite field.

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure()
sage: F.gen(2)
z2
sage: type(F.gen(2))
<class 'sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_pseudo_conway_with_category.element_class'>

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure()
>>> F.gen(Integer(2))
z2
>>> type(F.gen(Integer(2)))
<class 'sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_pseudo_conway_with_category.element_class'>

Sage Live

F = GF(3).algebraic_closure()
F.gen(2)
type(F.gen(2))

as_finite_field_element(minimal=False)[source]¶

Return self as a finite field element.

INPUT:

minimal – boolean (default: False); if True, always return the smallest subfield containing self

OUTPUT:

a triple (field, element, morphism) where field is a finite field, element an element of field and morphism a morphism from field to self.parent().

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure('t')
sage: t = F.gen(5)
sage: t.as_finite_field_element()
(Finite Field in t5 of size 3^5,
 t5,
 Ring morphism:
  From: Finite Field in t5 of size 3^5
  To:   Algebraic closure of Finite Field of size 3
  Defn: t5 |--> t5)

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure('t')
>>> t = F.gen(Integer(5))
>>> t.as_finite_field_element()
(Finite Field in t5 of size 3^5,
 t5,
 Ring morphism:
  From: Finite Field in t5 of size 3^5
  To:   Algebraic closure of Finite Field of size 3
  Defn: t5 |--> t5)

Sage Live

F = GF(3).algebraic_closure('t')
t = F.gen(5)
t.as_finite_field_element()

By default, field is not necessarily minimal. We can force it to be minimal using the minimal option:

Sage

sage: s = t + 1 - t
sage: s.as_finite_field_element()[0]
Finite Field in t5 of size 3^5
sage: s.as_finite_field_element(minimal=True)[0]
Finite Field of size 3

Python

>>> from sage.all import *
>>> s = t + Integer(1) - t
>>> s.as_finite_field_element()[Integer(0)]
Finite Field in t5 of size 3^5
>>> s.as_finite_field_element(minimal=True)[Integer(0)]
Finite Field of size 3

Sage Live

s = t + 1 - t
s.as_finite_field_element()[0]
s.as_finite_field_element(minimal=True)[0]

This also works when the element has to be converted between two non-trivial finite subfields (see Issue #16509):

Sage

sage: K = GF(5).algebraic_closure()
sage: z = K.gen(5) - K.gen(5) + K.gen(2)
sage: z.as_finite_field_element(minimal=True)
(Finite Field in z2 of size 5^2, z2, Ring morphism:
   From: Finite Field in z2 of size 5^2
   To:   Algebraic closure of Finite Field of size 5
   Defn: z2 |--> z2)

Python

>>> from sage.all import *
>>> K = GF(Integer(5)).algebraic_closure()
>>> z = K.gen(Integer(5)) - K.gen(Integer(5)) + K.gen(Integer(2))
>>> z.as_finite_field_element(minimal=True)
(Finite Field in z2 of size 5^2, z2, Ring morphism:
   From: Finite Field in z2 of size 5^2
   To:   Algebraic closure of Finite Field of size 5
   Defn: z2 |--> z2)

Sage Live

K = GF(5).algebraic_closure()
z = K.gen(5) - K.gen(5) + K.gen(2)
z.as_finite_field_element(minimal=True)

There are automatic coercions between the various subfields:

Sage

sage: a = K.gen(2) + 1
sage: _,b,_ = a.as_finite_field_element()
sage: K4 = K.subfield(4)[0]
sage: K4(b)
z4^3 + z4^2 + z4 + 4
sage: b.minimal_polynomial() == K4(b).minimal_polynomial()
True
sage: K(K4(b)) == K(b)
True

Python

>>> from sage.all import *
>>> a = K.gen(Integer(2)) + Integer(1)
>>> _,b,_ = a.as_finite_field_element()
>>> K4 = K.subfield(Integer(4))[Integer(0)]
>>> K4(b)
z4^3 + z4^2 + z4 + 4
>>> b.minimal_polynomial() == K4(b).minimal_polynomial()
True
>>> K(K4(b)) == K(b)
True

Sage Live

a = K.gen(2) + 1
_,b,_ = a.as_finite_field_element()
K4 = K.subfield(4)[0]
K4(b)
b.minimal_polynomial() == K4(b).minimal_polynomial()
K(K4(b)) == K(b)

You can also use the inclusions that are implemented at the level of the algebraic closure:

Sage

sage: f = K.inclusion(2,4); f
Ring morphism:
  From: Finite Field in z2 of size 5^2
  To:   Finite Field in z4 of size 5^4
  Defn: z2 |--> z4^3 + z4^2 + z4 + 3
sage: f(b)
z4^3 + z4^2 + z4 + 4

Python

>>> from sage.all import *
>>> f = K.inclusion(Integer(2),Integer(4)); f
Ring morphism:
  From: Finite Field in z2 of size 5^2
  To:   Finite Field in z4 of size 5^4
  Defn: z2 |--> z4^3 + z4^2 + z4 + 3
>>> f(b)
z4^3 + z4^2 + z4 + 4

Sage Live

f = K.inclusion(2,4); f
f(b)

change_level(n)[source]¶

Return a representation of self as an element of the subfield of degree \(n\) of the parent, if possible.

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure()
sage: z = F.gen(4)
sage: (z^10).change_level(6)
2*z6^5 + 2*z6^3 + z6^2 + 2*z6 + 2
sage: z.change_level(6)
Traceback (most recent call last):
...
ValueError: z4 is not in the image of Ring morphism:
  From: Finite Field in z2 of size 3^2
  To:   Finite Field in z4 of size 3^4
  Defn: z2 |--> 2*z4^3 + 2*z4^2 + 1

sage: a = F(1).change_level(3); a
1
sage: a.change_level(2)
1
sage: F.gen(3).change_level(1)
Traceback (most recent call last):
...
ValueError: z3 is not in the image of Ring morphism:
  From: Finite Field of size 3
  To:   Finite Field in z3 of size 3^3
  Defn: 1 |--> 1

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure()
>>> z = F.gen(Integer(4))
>>> (z**Integer(10)).change_level(Integer(6))
2*z6^5 + 2*z6^3 + z6^2 + 2*z6 + 2
>>> z.change_level(Integer(6))
Traceback (most recent call last):
...
ValueError: z4 is not in the image of Ring morphism:
  From: Finite Field in z2 of size 3^2
  To:   Finite Field in z4 of size 3^4
  Defn: z2 |--> 2*z4^3 + 2*z4^2 + 1

>>> a = F(Integer(1)).change_level(Integer(3)); a
1
>>> a.change_level(Integer(2))
1
>>> F.gen(Integer(3)).change_level(Integer(1))
Traceback (most recent call last):
...
ValueError: z3 is not in the image of Ring morphism:
  From: Finite Field of size 3
  To:   Finite Field in z3 of size 3^3
  Defn: 1 |--> 1

Sage Live

F = GF(3).algebraic_closure()
z = F.gen(4)
(z^10).change_level(6)
z.change_level(6)
a = F(1).change_level(3); a
a.change_level(2)
F.gen(3).change_level(1)

is_square()[source]¶

Return True if self is a square.

This always returns True.

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure()
sage: F.gen(2).is_square()
True

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure()
>>> F.gen(Integer(2)).is_square()
True

Sage Live

F = GF(3).algebraic_closure()
F.gen(2).is_square()

minimal_polynomial()[source]¶

Return the minimal polynomial of self over the prime field.

EXAMPLES:

Sage

sage: F = GF(11).algebraic_closure()
sage: F.gen(3).minpoly()
x^3 + 2*x + 9

Python

>>> from sage.all import *
>>> F = GF(Integer(11)).algebraic_closure()
>>> F.gen(Integer(3)).minpoly()
x^3 + 2*x + 9

Sage Live

F = GF(11).algebraic_closure()
F.gen(3).minpoly()

minpoly()[source]¶

Return the minimal polynomial of self over the prime field.

EXAMPLES:

Sage

sage: F = GF(11).algebraic_closure()
sage: F.gen(3).minpoly()
x^3 + 2*x + 9

Python

>>> from sage.all import *
>>> F = GF(Integer(11)).algebraic_closure()
>>> F.gen(Integer(3)).minpoly()
x^3 + 2*x + 9

Sage Live

F = GF(11).algebraic_closure()
F.gen(3).minpoly()

multiplicative_order()[source]¶

Return the multiplicative order of self.

EXAMPLES:

Sage

sage: K = GF(7).algebraic_closure()
sage: K.gen(5).multiplicative_order()
16806
sage: (K.gen(1) + K.gen(2) + K.gen(3)).multiplicative_order()
7353

Python

>>> from sage.all import *
>>> K = GF(Integer(7)).algebraic_closure()
>>> K.gen(Integer(5)).multiplicative_order()
16806
>>> (K.gen(Integer(1)) + K.gen(Integer(2)) + K.gen(Integer(3))).multiplicative_order()
7353

Sage Live

K = GF(7).algebraic_closure()
K.gen(5).multiplicative_order()
(K.gen(1) + K.gen(2) + K.gen(3)).multiplicative_order()

nth_root(n)[source]¶

Return an \(n\)-th root of self.

EXAMPLES:

Sage

sage: F = GF(5).algebraic_closure()
sage: t = F.gen(2) + 1
sage: s = t.nth_root(15); s
4*z6^5 + 3*z6^4 + 2*z6^3 + 2*z6^2 + 4
sage: s**15 == t
True

Python

>>> from sage.all import *
>>> F = GF(Integer(5)).algebraic_closure()
>>> t = F.gen(Integer(2)) + Integer(1)
>>> s = t.nth_root(Integer(15)); s
4*z6^5 + 3*z6^4 + 2*z6^3 + 2*z6^2 + 4
>>> s**Integer(15) == t
True

Sage Live

F = GF(5).algebraic_closure()
t = F.gen(2) + 1
s = t.nth_root(15); s
s**15 == t

Todo

This function could probably be made faster.

pth_power(k=1)[source]¶

Return the \(p^k\)-th power of self, where \(p\) is the characteristic of self.parent().

EXAMPLES:

Sage

sage: K = GF(13).algebraic_closure('t')
sage: t3 = K.gen(3)
sage: s = 1 + t3 + t3**2
sage: s.pth_power()
10*t3^2 + 6*t3
sage: s.pth_power(2)
2*t3^2 + 6*t3 + 11
sage: s.pth_power(3)
t3^2 + t3 + 1
sage: s.pth_power(3).parent() is K
True

Python

>>> from sage.all import *
>>> K = GF(Integer(13)).algebraic_closure('t')
>>> t3 = K.gen(Integer(3))
>>> s = Integer(1) + t3 + t3**Integer(2)
>>> s.pth_power()
10*t3^2 + 6*t3
>>> s.pth_power(Integer(2))
2*t3^2 + 6*t3 + 11
>>> s.pth_power(Integer(3))
t3^2 + t3 + 1
>>> s.pth_power(Integer(3)).parent() is K
True

Sage Live

K = GF(13).algebraic_closure('t')
t3 = K.gen(3)
s = 1 + t3 + t3**2
s.pth_power()
s.pth_power(2)
s.pth_power(3)
s.pth_power(3).parent() is K

pth_root(k=1)[source]¶

Return the unique \(p^k\)-th root of self, where \(p\) is the characteristic of self.parent().

EXAMPLES:

Sage

sage: K = GF(13).algebraic_closure('t')
sage: t3 = K.gen(3)
sage: s = 1 + t3 + t3**2
sage: s.pth_root()
2*t3^2 + 6*t3 + 11
sage: s.pth_root(2)
10*t3^2 + 6*t3
sage: s.pth_root(3)
t3^2 + t3 + 1
sage: s.pth_root(2).parent() is K
True

Python

>>> from sage.all import *
>>> K = GF(Integer(13)).algebraic_closure('t')
>>> t3 = K.gen(Integer(3))
>>> s = Integer(1) + t3 + t3**Integer(2)
>>> s.pth_root()
2*t3^2 + 6*t3 + 11
>>> s.pth_root(Integer(2))
10*t3^2 + 6*t3
>>> s.pth_root(Integer(3))
t3^2 + t3 + 1
>>> s.pth_root(Integer(2)).parent() is K
True

Sage Live

K = GF(13).algebraic_closure('t')
t3 = K.gen(3)
s = 1 + t3 + t3**2
s.pth_root()
s.pth_root(2)
s.pth_root(3)
s.pth_root(2).parent() is K

sqrt(all=False)[source]¶

Return a square root of self.

If the optional keyword argument all is set to True, return a list of all square roots of self instead.

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure()
sage: F.gen(2).sqrt()
z4^3 + z4 + 1
sage: F.gen(2).sqrt(all=True)
[z4^3 + z4 + 1, 2*z4^3 + 2*z4 + 2]
sage: (F.gen(2)^2).sqrt()
z2
sage: (F.gen(2)^2).sqrt(all=True)
[z2, 2*z2]

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure()
>>> F.gen(Integer(2)).sqrt()
z4^3 + z4 + 1
>>> F.gen(Integer(2)).sqrt(all=True)
[z4^3 + z4 + 1, 2*z4^3 + 2*z4 + 2]
>>> (F.gen(Integer(2))**Integer(2)).sqrt()
z2
>>> (F.gen(Integer(2))**Integer(2)).sqrt(all=True)
[z2, 2*z2]

Sage Live

F = GF(3).algebraic_closure()
F.gen(2).sqrt()
F.gen(2).sqrt(all=True)
(F.gen(2)^2).sqrt()
(F.gen(2)^2).sqrt(all=True)

class sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_generic(base_ring, name, category=None)[source]¶

Bases: Field

Algebraic closure of a finite field.

Element[source]¶: alias of AlgebraicClosureFiniteFieldElement

algebraic_closure()[source]¶

Return an algebraic closure of self.

This always returns self.

EXAMPLES:

Sage

sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
sage: F = AlgebraicClosureFiniteField(GF(5), 'z')
sage: F.algebraic_closure() is F
True

Python

>>> from sage.all import *
>>> from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
>>> F = AlgebraicClosureFiniteField(GF(Integer(5)), 'z')
>>> F.algebraic_closure() is F
True

Sage Live

from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
F = AlgebraicClosureFiniteField(GF(5), 'z')
F.algebraic_closure() is F

characteristic()[source]¶

Return the characteristic of self.

EXAMPLES:

Sage

sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
sage: p = next_prime(1000)
sage: F = AlgebraicClosureFiniteField(GF(p), 'z')
sage: F.characteristic() == p
True

Python

>>> from sage.all import *
>>> from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
>>> p = next_prime(Integer(1000))
>>> F = AlgebraicClosureFiniteField(GF(p), 'z')
>>> F.characteristic() == p
True

Sage Live

from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
p = next_prime(1000)
F = AlgebraicClosureFiniteField(GF(p), 'z')
F.characteristic() == p

gen(n)[source]¶

Return the \(n\)-th generator of self.

EXAMPLES:

Sage

sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
sage: F = AlgebraicClosureFiniteField(GF(5), 'z')
sage: F.gen(2)
z2

Python

>>> from sage.all import *
>>> from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
>>> F = AlgebraicClosureFiniteField(GF(Integer(5)), 'z')
>>> F.gen(Integer(2))
z2

Sage Live

from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
F = AlgebraicClosureFiniteField(GF(5), 'z')
F.gen(2)

gens()[source]¶

Return a family of generators of self.

OUTPUT:

a Family, indexed by the positive integers, whose \(n\)-th element is self.gen(n).

EXAMPLES:

Sage

sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
sage: F = AlgebraicClosureFiniteField(GF(5), 'z')
sage: g = F.gens(); g
Lazy family (...(i))_{i in Positive integers}
sage: g[3]
z3

Python

>>> from sage.all import *
>>> from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
>>> F = AlgebraicClosureFiniteField(GF(Integer(5)), 'z')
>>> g = F.gens(); g
Lazy family (...(i))_{i in Positive integers}
>>> g[Integer(3)]
z3

Sage Live

from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
F = AlgebraicClosureFiniteField(GF(5), 'z')
g = F.gens(); g
g[3]

inclusion(m, n)[source]¶

Return the canonical inclusion map from the subfield of degree \(m\) to the subfield of degree \(n\).

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure()
sage: F.inclusion(1, 2)
Ring morphism:
  From: Finite Field of size 3
  To:   Finite Field in z2 of size 3^2
  Defn: 1 |--> 1
sage: F.inclusion(2, 4)
Ring morphism:
  From: Finite Field in z2 of size 3^2
  To:   Finite Field in z4 of size 3^4
  Defn: z2 |--> 2*z4^3 + 2*z4^2 + 1

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure()
>>> F.inclusion(Integer(1), Integer(2))
Ring morphism:
  From: Finite Field of size 3
  To:   Finite Field in z2 of size 3^2
  Defn: 1 |--> 1
>>> F.inclusion(Integer(2), Integer(4))
Ring morphism:
  From: Finite Field in z2 of size 3^2
  To:   Finite Field in z4 of size 3^4
  Defn: z2 |--> 2*z4^3 + 2*z4^2 + 1

Sage Live

F = GF(3).algebraic_closure()
F.inclusion(1, 2)
F.inclusion(2, 4)

ngens()[source]¶

Return the number of generators of self, which is infinity.

EXAMPLES:

Sage

sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
sage: AlgebraicClosureFiniteField(GF(5), 'z').ngens()
+Infinity

Python

>>> from sage.all import *
>>> from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
>>> AlgebraicClosureFiniteField(GF(Integer(5)), 'z').ngens()
+Infinity

Sage Live

from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField
AlgebraicClosureFiniteField(GF(5), 'z').ngens()

some_elements()[source]¶

Return some elements of this field.

EXAMPLES:

Sage

sage: F = GF(7).algebraic_closure()
sage: F.some_elements()
(1, z2, z3 + 1)

Python

>>> from sage.all import *
>>> F = GF(Integer(7)).algebraic_closure()
>>> F.some_elements()
(1, z2, z3 + 1)

Sage Live

F = GF(7).algebraic_closure()
F.some_elements()

subfield(n)[source]¶

Return the unique subfield of degree \(n\) of self together with its canonical embedding into self.

EXAMPLES:

Sage

sage: F = GF(3).algebraic_closure()
sage: F.subfield(1)
(Finite Field of size 3,
 Ring morphism:
   From: Finite Field of size 3
   To:   Algebraic closure of Finite Field of size 3
   Defn: 1 |--> 1)
sage: F.subfield(4)
(Finite Field in z4 of size 3^4,
 Ring morphism:
   From: Finite Field in z4 of size 3^4
   To:   Algebraic closure of Finite Field of size 3
   Defn: z4 |--> z4)

Python

>>> from sage.all import *
>>> F = GF(Integer(3)).algebraic_closure()
>>> F.subfield(Integer(1))
(Finite Field of size 3,
 Ring morphism:
   From: Finite Field of size 3
   To:   Algebraic closure of Finite Field of size 3
   Defn: 1 |--> 1)
>>> F.subfield(Integer(4))
(Finite Field in z4 of size 3^4,
 Ring morphism:
   From: Finite Field in z4 of size 3^4
   To:   Algebraic closure of Finite Field of size 3
   Defn: z4 |--> z4)

Sage Live

F = GF(3).algebraic_closure()
F.subfield(1)
F.subfield(4)

class sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_pseudo_conway(base_ring, name, category=None, lattice=None, use_database=True)[source]¶

Bases: WithEqualityById, AlgebraicClosureFiniteField_generic

Algebraic closure of a finite field, constructed using pseudo-Conway polynomials.

EXAMPLES:

Sage

sage: F = GF(5).algebraic_closure(implementation='pseudo_conway')
sage: F.cardinality()
+Infinity
sage: F.algebraic_closure() is F
True
sage: x = F(3).nth_root(12); x
z4^3 + z4^2 + 4*z4
sage: x**12
3

Python

>>> from sage.all import *
>>> F = GF(Integer(5)).algebraic_closure(implementation='pseudo_conway')
>>> F.cardinality()
+Infinity
>>> F.algebraic_closure() is F
True
>>> x = F(Integer(3)).nth_root(Integer(12)); x
z4^3 + z4^2 + 4*z4
>>> x**Integer(12)
3

Sage Live

F = GF(5).algebraic_closure(implementation='pseudo_conway')
F.cardinality()
F.algebraic_closure() is F
x = F(3).nth_root(12); x
x**12