Base Classes for Finite Fields

AUTHORS:

  • Adrien Brochard, David Roe, Jeroen Demeyer, Julian Rueth, Niles Johnson, Peter Bruin, Travis Scrimshaw, Xavier Caruso: initial version
class sage.rings.finite_rings.finite_field_base.FiniteField

Bases: sage.rings.ring.Field

Abstract base class for finite fields.

algebraic_closure(name='z', **kwds)

Return an algebraic closure of self.

INPUT:

  • name – string (default: ‘z’): prefix to use for variable names of subfields
  • implementation – string (optional): specifies how to construct the algebraic closure. The only value supported at the moment is 'pseudo_conway'. For more details, see algebraic_closure_finite_field.

OUTPUT:

An algebraic closure of self. Note that mathematically speaking, this is only unique up to non-unique isomorphism. To obtain canonically defined algebraic closures, one needs an algorithm that also provides a canonical isomorphism between any two algebraic closures constructed using the algorithm.

This non-uniqueness problem can in principle be solved by using Conway polynomials; see for example Wikipedia article Conway_polynomial_(finite_fields). These have the drawback that computing them takes a long time. Therefore Sage implements a variant called pseudo-Conway polynomials, which are easier to compute but do not determine an algebraic closure up to unique isomorphism.

The output of this method is cached, so that within the same Sage session, calling it multiple times will return the same algebraic closure (i.e. the same Sage object). Despite this, the non-uniqueness of the current implementation means that coercion and pickling cannot work as one might expect. See below for an example.

EXAMPLES:

sage: F = GF(5).algebraic_closure()
sage: F
Algebraic closure of Finite Field of size 5
sage: F.gen(3)
z3

The default name is ‘z’ but you can change it through the option name:

sage: Ft = GF(5).algebraic_closure('t')
sage: Ft.gen(3)
t3

Because Sage currently only implements algebraic closures using a non-unique definition (see above), it is currently impossible to implement pickling in such a way that a pickled and unpickled element compares equal to the original:

sage: F = GF(7).algebraic_closure()
sage: x = F.gen(2)
sage: loads(dumps(x)) == x
False

Note

This is currently only implemented for prime fields.

cardinality()

Return the cardinality of self.

Same as order().

EXAMPLES:

sage: GF(997).cardinality()
997
construction()

Return the construction of this finite field, as a ConstructionFunctor and the base field.

EXAMPLES:

sage: v = GF(3^3).construction(); v
(AlgebraicExtensionFunctor, Finite Field of size 3)
sage: v[0].polys[0]
3
sage: v = GF(2^1000, 'a').construction(); v[0].polys[0]
a^1000 + a^5 + a^4 + a^3 + 1
dual_basis(basis=None, check=True)

Return the dual basis of basis, or the dual basis of the power basis if no basis is supplied.

If \(e = \{e_0, e_1, ..., e_{n-1}\}\) is a basis of \(\GF{p^n}\) as a vector space over \(\GF{p}\), then the dual basis of \(e\), \(d = \{d_0, d_1, ..., d_{n-1}\}\), is the unique basis such that \(\mathrm{Tr}(e_i d_j) = \delta_{i,j}, 0 \leq i,j \leq n-1\), where \(\mathrm{Tr}\) is the trace function.

INPUT:

  • basis – (default: None): a basis of the finite field self, \(\GF{p^n}\), as a vector space over the base field \(\GF{p}\). Uses the power basis \(\{x^i : 0 \leq i \leq n-1\}\) as input if no basis is supplied, where \(x\) is the generator of self.
  • check – (default: True): verifies that basis is a valid basis of self.

ALGORITHM:

The algorithm used to calculate the dual basis comes from pages 110–111 of [McE1987].

Let \(e = \{e_0, e_1, ..., e_{n-1}\}\) be a basis of \(\GF{p^n}\) as a vector space over \(\GF{p}\) and \(d = \{d_0, d_1, ..., d_{n-1}\}\) be the dual basis of \(e\). Since \(e\) is a basis, we can rewrite any \(d_c, 0 \leq c \leq n-1\), as \(d_c = \beta_0 e_0 + \beta_1 e_1 + ... + \beta_{n-1} e_{n-1}\), for some \(\beta_0, \beta_1, ..., \beta_{n-1} \in \GF{p}\). Using properties of the trace function, we can rewrite the \(n\) equations of the form \(\mathrm{Tr}(e_i d_c) = \delta_{i,c}\) and express the result as the matrix vector product: \(A [\beta_0, \beta_1, ..., \beta_{n-1}] = i_c\), where the \(i,j\)-th element of \(A\) is \(\mathrm{Tr(e_i e_j)}\) and \(i_c\) is the \(i\)-th column of the \(n \times n\) identity matrix. Since \(A\) is an invertible matrix, \([\beta_0, \beta_1, ..., \beta_{n-1}] = A^{-1} i_c\), from which we can easily calculate \(d_c\).

EXAMPLES:

sage: F.<a> = GF(2^4)
sage: F.dual_basis(basis=None, check=False)
[a^3 + 1, a^2, a, 1]

We can test that the dual basis returned satisfies the defining property of a dual basis: \(\mathrm{Tr}(e_i d_j) = \delta_{i,j}, 0 \leq i,j \leq n-1\)

sage: F.<a> = GF(7^4)
sage: e = [4*a^3, 2*a^3 + a^2 + 3*a + 5,
....:      3*a^3 + 5*a^2 + 4*a + 2, 2*a^3 + 2*a^2 + 2]
sage: d = F.dual_basis(e, check=True); d
[3*a^3 + 4*a^2 + 6*a + 2, a^3 + 6*a + 5,
3*a^3 + 6*a^2 + 2*a + 5, 5*a^2 + 4*a + 3]
sage: vals = [[(x * y).trace() for x in e] for y in d]
sage: matrix(vals) == matrix.identity(4)
True

We can test that if \(d\) is the dual basis of \(e\), then \(e\) is the dual basis of \(d\):

sage: F.<a> = GF(7^8)
sage: e = [a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7]
sage: d = F.dual_basis(e, check=False); d
[6*a^6 + 4*a^5 + 4*a^4 + a^3 + 6*a^2 + 3,
6*a^7 + 4*a^6 + 4*a^5 + 2*a^4 + a^2,
4*a^6 + 5*a^5 + 5*a^4 + 4*a^3 + 5*a^2 + a + 6,
5*a^7 + a^6 + a^4 + 4*a^3 + 4*a^2 + 1,
2*a^7 + 5*a^6 + a^5 + a^3 + 5*a^2 + 2*a + 4,
a^7 + 2*a^6 + 5*a^5 + a^4 + 5*a^2 + 4*a + 4,
a^7 + a^6 + 2*a^5 + 5*a^4 + a^3 + 4*a^2 + 4*a + 6,
5*a^7 + a^6 + a^5 + 2*a^4 + 5*a^3 + 6*a]
sage: F.dual_basis(d)
[1, a, a^2, a^3, a^4, a^5, a^6, a^7]

We cannot calculate the dual basis if basis is not a valid basis.

sage: F.<a> = GF(2^3)
sage: F.dual_basis([a], check=True)
Traceback (most recent call last):
...
ValueError: basis length should be 3, not 1

sage: F.dual_basis([a^0, a, a^0 + a], check=True)
Traceback (most recent call last):
...
ValueError: value of 'basis' keyword is not a basis

AUTHOR:

  • Thomas Gagne (2015-06-16)
extension(modulus, name=None, names=None, map=False, embedding=None, **kwds)

Return an extension of this finite field.

INPUT:

  • modulus – a polynomial with coefficients in self, or an integer.
  • name – string: the name of the generator in the new extension
  • map – boolean (default: False): if False, return just the extension \(E\); if True, return a pair \((E, f)\), where \(f\) is an embedding of self into \(E\).
  • embedding – currently not used; for compatibility with other AlgebraicExtensionFunctor calls.
  • **kwds: further keywords, passed to the finite field constructor.

OUTPUT:

An extension of the given modulus, or pseudo-Conway of the given degree if modulus is an integer.

EXAMPLES:

sage: k = GF(2)
sage: R.<x> = k[]
sage: k.extension(x^1000 + x^5 + x^4 + x^3 + 1, 'a')
Finite Field in a of size 2^1000
sage: k = GF(3^4)
sage: R.<x> = k[]
sage: k.extension(3)
Finite Field in z12 of size 3^12
sage: K = k.extension(2, 'a')
sage: k.is_subring(K)
True

An example using the map argument:

sage: F = GF(5)
sage: E, f = F.extension(2, 'b', map=True)
sage: E
Finite Field in b of size 5^2
sage: f
Ring morphism:
  From: Finite Field of size 5
  To:   Finite Field in b of size 5^2
  Defn: 1 |--> 1
sage: f.parent()
Set of field embeddings from Finite Field of size 5 to Finite Field in b of size 5^2

Extensions of non-prime finite fields by polynomials are not yet supported: we fall back to generic code:

sage: k.extension(x^5 + x^2 + x - 1)
Univariate Quotient Polynomial Ring in x over Finite Field in z4 of size 3^4 with modulus x^5 + x^2 + x + 2
factored_order()

Returns the factored order of this field. For compatibility with integer_mod_ring.

EXAMPLES:

sage: GF(7^2,'a').factored_order()
7^2
factored_unit_order()

Returns the factorization of self.order()-1, as a 1-tuple.

The format is for compatibility with integer_mod_ring.

EXAMPLES:

sage: GF(7^2,'a').factored_unit_order()
(2^4 * 3,)
frobenius_endomorphism(n=1)

INPUT:

  • n – an integer (default: 1)

OUTPUT:

The \(n\)-th power of the absolute arithmetic Frobenius endomorphism on this finite field.

EXAMPLES:

sage: k.<t> = GF(3^5)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^3 on Finite Field in t of size 3^5

sage: a = k.random_element()
sage: Frob(a) == a^3
True

We can specify a power:

sage: k.frobenius_endomorphism(2)
Frobenius endomorphism t |--> t^(3^2) on Finite Field in t of size 3^5

The result is simplified if possible:

sage: k.frobenius_endomorphism(6)
Frobenius endomorphism t |--> t^3 on Finite Field in t of size 3^5
sage: k.frobenius_endomorphism(5)
Identity endomorphism of Finite Field in t of size 3^5

Comparisons work:

sage: k.frobenius_endomorphism(6) == Frob
True
sage: from sage.categories.morphism import IdentityMorphism
sage: k.frobenius_endomorphism(5) == IdentityMorphism(k)
True

AUTHOR:

  • Xavier Caruso (2012-06-29)
gen()

Return a generator of this field (over its prime field). As this is an abstract base class, this just raises a NotImplementedError.

EXAMPLES:

sage: K = GF(17)
sage: sage.rings.finite_rings.finite_field_base.FiniteField.gen(K)
Traceback (most recent call last):
...
NotImplementedError
is_conway()

Return True if self is defined by a Conway polynomial.

EXAMPLES:

sage: GF(5^3, 'a').is_conway()
True
sage: GF(5^3, 'a', modulus='adleman-lenstra').is_conway()
False
sage: GF(next_prime(2^16, 2), 'a').is_conway()
False
is_field(proof=True)

Returns whether or not the finite field is a field, i.e., always returns True.

EXAMPLES:

sage: k.<a> = FiniteField(3^4)
sage: k.is_field()
True
is_perfect()

Return whether this field is perfect, i.e., every element has a \(p\)-th root. Always returns True since finite fields are perfect.

EXAMPLES:

sage: GF(2).is_perfect()
True
is_prime_field()

Return True if self is a prime field, i.e., has degree 1.

EXAMPLES:

sage: GF(3^7, 'a').is_prime_field()
False
sage: GF(3, 'a').is_prime_field()
True
modulus()

Return the minimal polynomial of the generator of self over the prime finite field.

The minimal polynomial of an element \(a\) in a field is the unique monic irreducible polynomial of smallest degree with coefficients in the base field that has \(a\) as a root. In finite field extensions, \(\GF{p^n}\), the base field is \(\GF{p}\).

OUTPUT:

  • a monic polynomial over \(\GF{p}\) in the variable \(x\).

EXAMPLES:

sage: F.<a> = GF(7^2); F
Finite Field in a of size 7^2
sage: F.polynomial_ring()
Univariate Polynomial Ring in a over Finite Field of size 7
sage: f = F.modulus(); f
x^2 + 6*x + 3
sage: f(a)
0

Although \(f\) is irreducible over the base field, we can double-check whether or not \(f\) factors in \(F\) as follows. The command F['x'](f) coerces \(f\) as a polynomial with coefficients in \(F\). (Instead of a polynomial with coefficients over the base field.)

sage: f.factor()
x^2 + 6*x + 3
sage: F['x'](f).factor()
(x + a + 6) * (x + 6*a)

Here is an example with a degree 3 extension:

sage: G.<b> = GF(7^3); G
Finite Field in b of size 7^3
sage: g = G.modulus(); g
x^3 + 6*x^2 + 4
sage: g.degree(); G.degree()
3
3

For prime fields, this returns \(x - 1\) unless a custom modulus was given when constructing this field:

sage: k = GF(199)
sage: k.modulus()
x + 198
sage: var('x')
x
sage: k = GF(199, modulus=x+1)
sage: k.modulus()
x + 1

The given modulus is always made monic:

sage: k.<a> = GF(7^2, modulus=2*x^2-3, impl="pari_ffelt")
sage: k.modulus()
x^2 + 2
multiplicative_generator()

Return a primitive element of this finite field, i.e. a generator of the multiplicative group.

You can use multiplicative_generator() or primitive_element(), these mean the same thing.

Warning

This generator might change from one version of Sage to another.

EXAMPLES:

sage: k = GF(997)
sage: k.multiplicative_generator()
7
sage: k.<a> = GF(11^3)
sage: k.primitive_element()
a
sage: k.<b> = GF(19^32)
sage: k.multiplicative_generator()
b + 4
ngens()

The number of generators of the finite field. Always 1.

EXAMPLES:

sage: k = FiniteField(3^4, 'b')
sage: k.ngens()
1
order()

Return the order of this finite field.

EXAMPLES:

sage: GF(997).order()
997
polynomial(name=None)

Return the minimal polynomial of the generator of self over the prime finite field.

INPUT:

  • name – a variable name to use for the polynomial. By default, use the name given when constructing this field.

OUTPUT:

  • a monic polynomial over \(\GF{p}\) in the variable name.

See also

Except for the name argument, this is identical to the modulus() method.

EXAMPLES:

sage: k.<a> = FiniteField(9)
sage: k.polynomial('x')
x^2 + 2*x + 2
sage: k.polynomial()
a^2 + 2*a + 2

sage: F = FiniteField(9, 'a', impl='pari_ffelt')
sage: F.polynomial()
a^2 + 2*a + 2

sage: F = FiniteField(7^20, 'a', impl='pari_ffelt')
sage: f = F.polynomial(); f
a^20 + a^12 + 6*a^11 + 2*a^10 + 5*a^9 + 2*a^8 + 3*a^7 + a^6 + 3*a^5 + 3*a^3 + a + 3
sage: f(F.gen())
0

sage: k.<a> = GF(2^20, impl='ntl')
sage: k.polynomial()
a^20 + a^10 + a^9 + a^7 + a^6 + a^5 + a^4 + a + 1
sage: k.polynomial('FOO')
FOO^20 + FOO^10 + FOO^9 + FOO^7 + FOO^6 + FOO^5 + FOO^4 + FOO + 1
sage: a^20
a^10 + a^9 + a^7 + a^6 + a^5 + a^4 + a + 1
polynomial_ring(variable_name=None)

Returns the polynomial ring over the prime subfield in the same variable as this finite field.

EXAMPLES:

sage: k.<alpha> = FiniteField(3^4)
sage: k.polynomial_ring()
Univariate Polynomial Ring in alpha over Finite Field of size 3
primitive_element()

Return a primitive element of this finite field, i.e. a generator of the multiplicative group.

You can use multiplicative_generator() or primitive_element(), these mean the same thing.

Warning

This generator might change from one version of Sage to another.

EXAMPLES:

sage: k = GF(997)
sage: k.multiplicative_generator()
7
sage: k.<a> = GF(11^3)
sage: k.primitive_element()
a
sage: k.<b> = GF(19^32)
sage: k.multiplicative_generator()
b + 4
random_element(*args, **kwds)

A random element of the finite field. Passes arguments to random_element() function of underlying vector space.

EXAMPLES:

sage: k = GF(19^4, 'a')
sage: k.random_element()
a^3 + 3*a^2 + 6*a + 9

Passes extra positional or keyword arguments through:

sage: k.random_element(prob=0)
0
some_elements()

Returns a collection of elements of this finite field for use in unit testing.

EXAMPLES:

sage: k = GF(2^8,'a')
sage: k.some_elements() # random output
[a^4 + a^3 + 1, a^6 + a^4 + a^3, a^5 + a^4 + a, a^2 + a]
subfields(degree=0, name=None)

Return all subfields of self of the given degree, or all possible degrees if degree is \(0\).

The subfields are returned as absolute fields together with an embedding into self.

INPUT:

  • degree – (default: \(0\)) an integer
  • name – a string, a dictionary or None:
    • If degree is nonzero, then name must be a string (or None, if this is a pseudo-Conway extension), and will be the variable name of the returned field.
    • If degree is zero, the dictionary should have keys the divisors of the degree of this field, with the desired variable name for the field of that degree as an entry.
    • As a shortcut, you can provide a string and the degree of each subfield will be appended for the variable name of that subfield.
    • If None, uses the prefix of this field.

OUTPUT:

A list of pairs (K, e), where K ranges over the subfields of this field and e gives an embedding of K into self.

EXAMPLES:

sage: k = GF(2^21)
sage: k.subfields()
[(Finite Field of size 2,
  Ring morphism:
      From: Finite Field of size 2
      To:   Finite Field in z21 of size 2^21
      Defn: 1 |--> 1),
 (Finite Field in z3 of size 2^3,
  Ring morphism:
      From: Finite Field in z3 of size 2^3
      To:   Finite Field in z21 of size 2^21
      Defn: z3 |--> z21^20 + z21^19 + z21^17 + z21^15 + z21^11 + z21^9 + z21^8 + z21^6 + z21^2),
 (Finite Field in z7 of size 2^7,
  Ring morphism:
      From: Finite Field in z7 of size 2^7
      To:   Finite Field in z21 of size 2^21
      Defn: z7 |--> z21^20 + z21^19 + z21^17 + z21^15 + z21^14 + z21^6 + z21^4 + z21^3 + z21),
 (Finite Field in z21 of size 2^21,
  Ring morphism:
      From: Finite Field in z21 of size 2^21
      To:   Finite Field in z21 of size 2^21
      Defn: z21 |--> z21)]
unit_group_exponent()

The exponent of the unit group of the finite field. For a finite field, this is always the order minus 1.

EXAMPLES:

sage: k = GF(2^10, 'a')
sage: k.order()
1024
sage: k.unit_group_exponent()
1023
vector_space(subfield=None, basis=None, map=False)

Return the vector space over the subfield isomorphic to this finite field as a vector space, along with the isomorphisms.

INPUT:

  • subfield – a subfield of or a morphism into this finite field. If not given, the prime subfield is assumed. A subfield means a finite field with coercion to this finite field.
  • basis – a basis of the finite field as a vector space over the subfield. If not given, one is chosen automatically.
  • map – boolean (default: False); if True, isomorphisms from and to the vector space are also returned.

The basis maps to the standard basis of the vector space by the isomorphisms.

OUTPUT: if map is False,

  • vector space over the subfield or the domain of the morphism, isomorphic to this finite field.

and if map is True, then also

  • an isomorphism from the vector space to the finite field.
  • the inverse isomorphism to the vector space from the finite field.

EXAMPLES:

sage: GF(27,'a').vector_space()
Vector space of dimension 3 over Finite Field of size 3

sage: F = GF(8)
sage: E = GF(64)
sage: V, from_V, to_V = E.vector_space(F, map=True)
sage: V
Vector space of dimension 2 over Finite Field in z3 of size 2^3
sage: to_V(E.gen())
(0, 1)
sage: all(from_V(to_V(e)) == e for e in E)
True
sage: all(to_V(e1 + e2) == to_V(e1) + to_V(e2) for e1 in E for e2 in E)
True
sage: all(to_V(c * e) == c * to_V(e) for e in E for c in F)
True

sage: basis = [E.gen(), E.gen() + 1]
sage: W, from_W, to_W = E.vector_space(F, basis, map=True)
sage: all(from_W(to_W(e)) == e for e in E)
True
sage: all(to_W(c * e) == c * to_W(e) for e in E for c in F)
True
sage: all(to_W(e1 + e2) == to_W(e1) + to_W(e2) for e1 in E for e2 in E)  # long time
True
sage: to_W(basis[0]); to_W(basis[1])
(1, 0)
(0, 1)

sage: F = GF(9, 't', modulus=(x^2+x-1))
sage: E = GF(81)
sage: h = Hom(F,E).an_element()
sage: V, from_V, to_V = E.vector_space(h, map=True)
sage: V
Vector space of dimension 2 over Finite Field in t of size 3^2
sage: V.base_ring() is F
True
sage: all(from_V(to_V(e)) == e for e in E)
True
sage: all(to_V(e1 + e2) == to_V(e1) + to_V(e2) for e1 in E for e2 in E)
True
sage: all(to_V(h(c) * e) == c * to_V(e) for e in E for c in F)
True
zeta(n=None)

Return an element of multiplicative order n in this finite field. If there is no such element, raise ValueError.

Warning

In general, this returns an arbitrary element of the correct order. There are no compatibility guarantees: F.zeta(9)^3 may not be equal to F.zeta(3).

EXAMPLES:

sage: k = GF(7)
sage: k.zeta()
3
sage: k.zeta().multiplicative_order()
6
sage: k.zeta(3)
2
sage: k.zeta(3).multiplicative_order()
3
sage: k = GF(49, 'a')
sage: k.zeta().multiplicative_order()
48
sage: k.zeta(6)
3
sage: k.zeta(5)
Traceback (most recent call last):
...
ValueError: no 5th root of unity in Finite Field in a of size 7^2

Even more examples:

sage: GF(9,'a').zeta_order()
8
sage: GF(9,'a').zeta()
a
sage: GF(9,'a').zeta(4)
a + 1
sage: GF(9,'a').zeta()^2
a + 1

This works even in very large finite fields, provided that n can be factored (see trac ticket #25203):

sage: k.<a> = GF(2^2000)
sage: p = 8877945148742945001146041439025147034098690503591013177336356694416517527310181938001
sage: z = k.zeta(p)
sage: z
a^1999 + a^1996 + a^1995 + a^1994 + ... + a^7 + a^5 + a^4 + 1
sage: z ^ p
1
zeta_order()

Return the order of the distinguished root of unity in self.

EXAMPLES:

sage: GF(9,'a').zeta_order()
8
sage: GF(9,'a').zeta()
a
sage: GF(9,'a').zeta().multiplicative_order()
8
sage.rings.finite_rings.finite_field_base.is_FiniteField(x)

Return True if x is of type finite field, and False otherwise.

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_base import is_FiniteField
sage: is_FiniteField(GF(9,'a'))
True
sage: is_FiniteField(GF(next_prime(10^10)))
True

Note that the integers modulo n are not of type finite field, so this function returns False:

sage: is_FiniteField(Integers(7))
False
sage.rings.finite_rings.finite_field_base.unpickle_FiniteField_ext(_type, order, variable_name, modulus, kwargs)

Used to unpickle extensions of finite fields. Now superseded (hence no doctest), but kept around for backward compatibility.

sage.rings.finite_rings.finite_field_base.unpickle_FiniteField_prm(_type, order, variable_name, kwargs)

Used to unpickle finite prime fields. Now superseded (hence no doctest), but kept around for backward compatibility.