Module finite_field

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) defining polynomial for field, i.e.,
               generator of the field will be a root of this
               polynomial; if not specified the choice of
               definining polynomials can be arbitrary.
    elem_cache -- cache all elements to avoid creation time  (default: order<500)
    check_irreducible -- verify that the polynomial modulus is irreducible
    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

    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_modn_dense_ntl.Polynomial_dense_mod_p'>

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 accidently fool the constructor into thinking the
modulus is irreducible when it isn't mod p, since it actually
tests irreducibility modulo p.

    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

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!

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

For example, 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

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

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

Returns True if x is a prime finite field.

EXAMPLES:
    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

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 \code{ConwayPolynomials()} object, which is a table of
Conway polynomials.   For example, if \code{c=ConwayPolynomials}, then
\code{c.primes()} is a list of all primes for which the polynomials are
known, and for a given prime $p$,  \code{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.

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 \code{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

cache

Value:

{(2, None, None, '[(), {}]'): <weakref at 0x4bc4158; to 'FiniteField_p rime_modn' at 0x4b05750>}