AUTHORS:
Bases: sage.rings.finite_rings.finite_field_base.FiniteField
Finite Field of order , where
is a prime power (not a prime),
implemented using PARI POLMOD. This implementation is the default
implementation for
.
INPUT:
OUTPUT:
A finite field of order with the given variable name
EXAMPLES:
sage: P.<x> = PolynomialRing(GF(3))
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(9, 'a', modulus=(x^2 + 2*x + 2))
doctest:...: DeprecationWarning: The "pari_mod" finite field implementation is deprecated
See http://trac.sagemath.org/17297 for details.
sage: k
Finite Field in a of size 3^2
sage: k.is_field()
True
sage: k.characteristic()
3
sage: a = k.gen()
sage: a
a
sage: a.parent()
Finite Field in a of size 3^2
sage: a.charpoly('x')
x^2 + 2*x + 2
sage: [a^i for i in range(8)]
[1, a, a + 1, 2*a + 1, 2, 2*a, 2*a + 2, a + 2]
Fields can be coerced into sets or list and iterated over:
sage: list(k)
[0, 1, 2, a, a + 1, a + 2, 2*a, 2*a + 1, 2*a + 2]
The following is a native Python set:
sage: set(k)
{0, 1, 2, a, 2*a, a + 1, 2*a + 1, a + 2, 2*a + 2}
And the following is a Sage set:
sage: Set(k)
{0, 1, 2, a, a + 1, a + 2, 2*a, 2*a + 1, 2*a + 2}
We can also make a list via comprehension:
sage: [x for x in k]
[0, 1, 2, a, a + 1, a + 2, 2*a, 2*a + 1, 2*a + 2]
Next we compute with the finite field of order 16, where the name is named b:
sage: P.<x> = PolynomialRing(GF(2))
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k16 = FiniteField_ext_pari(16, "b", modulus=(x^4 + x + 1))
sage: z = k16.gen()
sage: z
b
sage: z.charpoly('x')
x^4 + x + 1
sage: k16.is_field()
True
sage: k16.characteristic()
2
sage: z.multiplicative_order()
15
Of course one can also make prime finite fields:
sage: k = FiniteField(7)
Note that the generator is 1:
sage: k.gen()
1
sage: k.gen().multiplicative_order()
1
Prime finite fields are implemented elsewhere, they cannot be constructed using FiniteField_ext_pari:
sage: k = FiniteField_ext_pari(7, 'a', modulus=polygen(GF(7)))
Traceback (most recent call last):
...
ValueError: The size of the finite field must not be prime.
Illustration of dumping and loading:
sage: K = FiniteField(7)
sage: loads(K.dumps()) == K
True
sage: K = FiniteField(7^10, 'b', impl='pari_mod')
doctest:...: DeprecationWarning: The "pari_mod" finite field implementation is deprecated
See http://trac.sagemath.org/17297 for details.
sage: loads(K.dumps()) == K
True
sage: K = FiniteField(7^10, 'a', impl='pari_mod')
sage: loads(K.dumps()) == K
True
In this example is large enough that Conway polynomials are not
used. Note that when the field is dumped the defining polynomial
is also dumped. Since
is determined by a random algorithm, it’s
important that
is dumped as part of
. If you quit Sage and
restart and remake a finite field of the same order (and the order is
large enough so that there is no Conway polynomial), then defining
polynomial is probably different. However, if you load a previously
saved field, that will have the same defining polynomial.
sage: K = GF(10007^10, 'a', impl='pari_mod')
sage: loads(K.dumps()) == K
True
Note
We do NOT yet define natural consistent inclusion maps between different finite fields.
Returns the characteristic of the finite field, which is a prime number.
EXAMPLES:
sage: k = FiniteField(3^4, 'a', impl='pari_mod')
sage: k.characteristic()
3
Returns the degree of the finite field, which is a positive integer.
EXAMPLES:
sage: FiniteField(3^20, 'a', impl='pari_mod').degree()
20
Return a generator of self over its prime field, which is a root of self.modulus().
INPUT:
OUTPUT:
An element of self such that self.modulus()(a) == 0.
Warning
This generator is not guaranteed to be a generator for the multiplicative group. To obtain the latter, use multiplicative_generator() or use the modulus="primitive" option when constructing the field.
EXAMPLES:
sage: FiniteField(2^4, "b", impl='pari_mod').gen()
b
sage: k = FiniteField(3^4, "alpha", impl='pari_mod')
sage: a = k.gen()
sage: a
alpha
sage: a^4
alpha^3 + 1
The number of elements of the finite field.
EXAMPLES:
sage: k = FiniteField(2^10, 'a', impl='pari_mod')
sage: k
Finite Field in a of size 2^10
sage: k.order()
1024