# Finite Fields¶

Sage supports arithmetic in finite prime and extension fields. Several implementation for prime fields are implemented natively in Sage for several sizes of primes $$p$$. These implementations are

• sage.rings.finite_rings.integer_mod.IntegerMod_int,
• sage.rings.finite_rings.integer_mod.IntegerMod_int64, and
• sage.rings.finite_rings.integer_mod.IntegerMod_gmp.

Small extension fields of cardinality $$< 2^{16}$$ are implemented using tables of Zech logs via the Givaro C++ library (sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro). While this representation is very fast it is limited to finite fields of small cardinality. Larger finite extension fields of order $$q >= 2^{16}$$ are internally represented as polynomials over smaller finite prime fields. If the characteristic of such a field is 2 then NTL is used internally to represent the field (sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e). In all other case the PARI C library is used (sage.rings.finite_rings.finite_field_ext_pari.FiniteField_ext_pari).

However, this distinction is internal only and the user usually does not have to worry about it because consistency across all implementations is aimed for. In all extension field implementations the user may either specify a minimal polynomial or leave the choice to Sage.

For small finite fields the default choice are Conway polynomials.

The Conway polynomial $$C_n$$ is the lexicographically first monic irreducible, primitive polynomial of degree $$n$$ over $$GF(p)$$ with the property that for a root $$\alpha$$ of $$C_n$$ we have that $$\beta= \alpha^{(p^n - 1)/(p^m - 1)}$$ is a root of $$C_m$$ for all $$m$$ dividing $$n$$. Sage contains a database of Conway polynomials which also can be queried independently of finite field construction.

While Sage supports basic arithmetic in finite fields some more advanced features for computing with finite fields are still not implemented. For instance, Sage does not calculate embeddings of finite fields yet.

EXAMPLES:

sage: k = GF(5); type(k)
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>

sage: k = GF(5^2,'c'); type(k)
<class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>

sage: k = GF(2^16,'c'); type(k)
<class 'sage.rings.finite_rings.finite_field_ntl_gf2e.FiniteField_ntl_gf2e_with_category'>

sage: k = GF(3^16,'c'); type(k)
<class 'sage.rings.finite_rings.finite_field_ext_pari.FiniteField_ext_pari_with_category'>


Finite Fields support iteration, starting with 0.

sage: k = GF(9, 'a')
sage: for i,x in enumerate(k):  print i,x
0 0
1 a
2 a + 1
3 2*a + 1
4 2
5 2*a
6 2*a + 2
7 a + 2
8 1
sage: for a in GF(5):
...    print a
0
1
2
3
4


We output the base rings of several finite fields.

sage: k = GF(3); type(k)
<class 'sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn_with_category'>
sage: k.base_ring()
Finite Field of size 3

sage: k = GF(9,'alpha'); type(k)
<class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>
sage: k.base_ring()
Finite Field of size 3

sage: k = GF(3^40,'b'); type(k)
<class 'sage.rings.finite_rings.finite_field_ext_pari.FiniteField_ext_pari_with_category'>
sage: k.base_ring()
Finite Field of size 3


Further examples:

sage: GF(2).is_field()
True
sage: GF(next_prime(10^20)).is_field()
True
sage: GF(19^20,'a').is_field()
True
sage: GF(8,'a').is_field()
True


AUTHORS:

• William Stein: initial version
• Robert Bradshaw: prime field implementation
• Martin Albrecht: Givaro and ntl.GF2E implementations
class sage.rings.finite_rings.constructor.FiniteFieldFactory

Return the globally unique finite field of given order with generator labeled by the given name and possibly with given modulus.

INPUT:

• order - int

• name - string; must be specified if not a prime field

• modulus - (optional) either a defining polynomial for the field, i.e., generator of the field will be a root of this polynomial; or a string:

• ‘conway’: force the use of a Conway polynomial, will raise a RuntimeError if none is found in the database;
• ‘random’: use a random irreducible polynomial;
• ‘default’: a Conway polynomial is used if found. Otherwise a sparse polynomial is used for binary fields and a random polynomial is used for other characteristics.

Other options might be available depending on the implementation.

• elem_cache - cache all elements to avoid creation time (default: order < 500)

• check_irreducible - verify that the polynomial modulus is irreducible

• proof – bool (default: True); if True use provable primality test; otherwise only use pseudoprimality test.

• args - additional parameters passed to finite field implementations

• kwds - additional keyword parameters passed to finite field implementations

ALIAS: You can also use GF instead of FiniteField - they are identical.

EXAMPLES:

sage: k.<a> = FiniteField(9); k
Finite Field in a of size 3^2
sage: parent(a)
Finite Field in a of size 3^2
sage: charpoly(a, 'y')
y^2 + 2*y + 2


We illustrate the proof flag. The following example would hang for a very long time if we didn’t use proof=False. (NOTE: Magma only supports proof=False for making finite fields, so falsely appears to be faster than Sage – see Trac 10975.):

sage: k = FiniteField(10^1000 + 453, proof=False)
sage: k = FiniteField((10^1000 + 453)^2, 'a', proof=False)      # long time -- about 5 seconds

sage: F.<x> = GF(5)[]
sage: K.<a> = GF(5**5, name='a', modulus=x^5 - x +1 )
sage: f = K.modulus(); f
x^5 + 4*x + 1
sage: type(f)
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>


The modulus must be irreducible:

sage: K.<a> = GF(5**5, name='a', modulus=x^5 - x )
Traceback (most recent call last):
...
ValueError: finite field modulus must be irreducible but it is not.


You can’t accidentally fool the constructor into thinking the modulus is irreducible when it isn’t mod p, since it actually tests irreducibility modulo p. Also, the modulus has to be of the right degree.

sage: F.<x> = QQ[]
sage: factor(x^5 + 2)
x^5 + 2
sage: K.<a> = GF(5**5, name='a', modulus=x^5 + 2 )
Traceback (most recent call last):
...
ValueError: finite field modulus must be irreducible but it is not.
sage: K.<a> = GF(5**5, name='a', modulus=x^3 + 3*x + 3)
Traceback (most recent call last):
...
ValueError: The degree of the modulus does not correspond to the
cardinality of the field.


If you wish to live dangerously, you can tell the constructor not to test irreducibility using check_irreducible=False, but this can easily lead to crashes and hangs - so do not do it unless you know that the modulus really is irreducible and has the correct degree!

sage: F.<x> = GF(5)[]
sage: K.<a> = GF(5**2, name='a', modulus=x^2 + 2, check_irreducible=False)

sage: L = GF(3**2, name='a', modulus=QQ[x](x - 1), check_irreducible=False)
sage: L.list()  # random
[0, a, 1, 2, 1, 2, 1, 2, 1]


The order of a finite field must be a prime power:

sage: GF(1)
Traceback (most recent call last):
...
ValueError: the order of a finite field must be > 1.
sage: GF(100)
Traceback (most recent call last):
...
ValueError: the order of a finite field must be a prime power.


Finite fields with explicit random modulus are not cached:

sage: k.<a> = GF(5**10, modulus='random')
sage: n.<a> = GF(5**10, modulus='random')
sage: n is k
False
sage: GF(5**10, 'a') is GF(5**10, 'a')
True


We check that various ways of creating the same finite field yield the same object, which is cached.

sage: K = GF(7, 'a')
sage: L = GF(7, 'b')
sage: K is L
True
sage: K = GF(4,'a'); K.modulus()
x^2 + x + 1
sage: L = GF(4,'a', K.modulus())
sage: K is L
True
sage: M = GF(4,'a', K.modulus().change_variable_name('y'))
sage: K is M
True


You may print finite field elements as integers. This currently only works if the order of field is $$<2^{16}$$, though.

sage: k.<a> = GF(2^8, repr='int')
sage: a
2

create_key_and_extra_args(order, name=None, modulus=None, names=None, impl=None, proof=None, **kwds)

EXAMPLES:

sage: GF.create_key_and_extra_args(9, 'a')
((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {})
sage: GF.create_key_and_extra_args(9, 'a', foo='value')
((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})

create_object(version, key, check_irreducible=True, elem_cache=None, names=None, **kwds)

EXAMPLES:

sage: K = GF(19) # indirect doctest
sage: TestSuite(K).run()

other_keys(key, K)

EXAMPLES:

sage: key, extra = GF.create_key_and_extra_args(9, 'a'); key
(9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True)
sage: K = GF.create_object(0, key); K
Finite Field in a of size 3^2
sage: GF.other_keys(key, K)
[(9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True),
(9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True)]

sage.rings.finite_rings.constructor.conway_polynomial(p, n)

Return the Conway polynomial of degree n over GF(p), which is loaded from a table.

If the requested polynomial is not known, this function raises a RuntimeError exception.

INPUT:

• p - int
• n - int

OUTPUT:

• Polynomial - a polynomial over the prime finite field GF(p).

Note

The first time this function is called a table is read from disk, which takes a fraction of a second. Subsequent calls do not require reloading the table.

See also the ConwayPolynomials() object, which is a table of Conway polynomials. For example, if c=ConwayPolynomials(), then c.primes() is a list of all primes for which the polynomials are known, and for a given prime $$p$$, c.degree(p) is a list of all degrees for which the Conway polynomials are known.

EXAMPLES:

sage: conway_polynomial(2,5)
x^5 + x^2 + 1
sage: conway_polynomial(101,5)
x^5 + 2*x + 99
sage: conway_polynomial(97,101)
Traceback (most recent call last):
...
RuntimeError: requested conway polynomial not in database.

sage.rings.finite_rings.constructor.exists_conway_polynomial(p, n)

Return True if the Conway polynomial over $$F_p$$ of degree $$n$$ is in the database and False otherwise.

If the Conway polynomial is in the database, to obtain it use the command conway_polynomial(p,n).

EXAMPLES:

sage: exists_conway_polynomial(2,3)
True
sage: exists_conway_polynomial(2,-1)
False
sage: exists_conway_polynomial(97,200)
False
sage: exists_conway_polynomial(6,6)
False

sage.rings.finite_rings.constructor.is_PrimeFiniteField(x)

Returns True if x is a prime finite field.

EXAMPLES:

sage: from sage.rings.finite_rings.constructor import is_PrimeFiniteField
sage: is_PrimeFiniteField(QQ)
False
sage: is_PrimeFiniteField(GF(7))
True
sage: is_PrimeFiniteField(GF(7^2,'a'))
False
sage: is_PrimeFiniteField(GF(next_prime(10^90,proof=False)))
True


Rational Numbers

#### Next topic

Base class for finite field elements