# Finite Extension Fields implemented via PARI.¶

AUTHORS:

• William Stein: initial version
• Jeroen Demeyer (2010-12-16): fix formatting of docstrings (trac ticket #10487)
class sage.rings.finite_rings.finite_field_ext_pari.FiniteField_ext_pari(q, name, modulus=None)

Finite Field of order $$q$$, where $$q$$ is a prime power (not a prime), implemented using PARI POLMOD. This implementation is the default implementation for $$q \geq 2^{16}$$.

INPUT:

• q – integer, size of the finite field, not prime
• name – variable used for printing element of the finite field. Also, two finite fields are considered equal if they have the same variable name, and not otherwise.
• modulus – you may provide a polynomial to use for reduction 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 random polynomial is used

OUTPUT:

A finite field of order $$q$$ with the given variable name

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(9, 'a')
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)
set([0, 1, 2, a, a + 1, a + 2, 2*a, 2*a + 1, 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: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k16 = FiniteField_ext_pari(16, "b")
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')
Traceback (most recent call last):
...
ValueError: The size of the finite field must not be prime.


sage: K = FiniteField(7)
True
sage: K = FiniteField(7^10, 'b', impl='pari_mod')
True
sage: K = FiniteField(7^10, 'a', impl='pari_mod')
True


In this example $$K$$ is large enough that Conway polynomials are not used. Note that when the field is dumped the defining polynomial $$f$$ is also dumped. Since $$f$$ is determined by a random algorithm, it’s important that $$f$$ is dumped as part of $$K$$. 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')
True


Note

We do NOT yet define natural consistent inclusion maps between different finite fields.

characteristic()

Returns the characteristic of the finite field, which is a prime number.

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(3^4, 'a')
sage: k.characteristic()
3

degree()

Returns the degree of the finite field, which is a positive integer.

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: FiniteField_ext_pari(3^20, 'a').degree()
20

gen(n=0)

Return a generator of the finite field.

This generator is a root of the defining polynomial of the finite field, and might differ between different runs or different architectures.

Warning

This generator is not guaranteed to be a generator for the multiplicative group. To obtain the latter, use multiplicative_generator().

INPUT:

• n – ignored

OUTPUT:

Field generator of finite field

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: FiniteField_ext_pari(2^4, "b").gen()
b
sage: k = FiniteField_ext_pari(3^4, "alpha")
sage: a = k.gen()
sage: a
alpha
sage: a^4
alpha^3 + 1

modulus()

Return the minimal polynomial of the generator of self in self.polynomial_ring('x').

EXAMPLES:

sage: F.<a> = GF(7^20, 'a', impl='pari_mod')
sage: f = F.modulus(); f
x^20 + x^12 + 6*x^11 + 2*x^10 + 5*x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^3 + x + 3

sage: f(a)
0

order()

The number of elements of the finite field.

EXAMPLES:

sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(2^10,'a')
sage: k
Finite Field in a of size 2^10
sage: k.order()
1024

polynomial(name=None)

Return the irreducible characteristic polynomial of the generator of this finite field, i.e., the polynomial $$f(x)$$ so elements of the finite field as elements modulo $$f$$.

EXAMPLES:

sage: k = FiniteField(17)
sage: k.polynomial('x')
x
sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
sage: k = FiniteField_ext_pari(9,'a')
sage: k.polynomial('x')
x^2 + 2*x + 2


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