AUTHORS:
TESTS:
sage: k = GF(3)
sage: TestSuite(k).run()
Bases: sage.rings.finite_rings.finite_field_base.FiniteField, sage.rings.finite_rings.integer_mod_ring.IntegerModRing_generic
Finite field of order where
is prime.
EXAMPLES:
sage: FiniteField(3)
Finite Field of size 3
sage: FiniteField(next_prime(1000))
Finite Field of size 1009
Return the characteristic of code{self}.
EXAMPLES:
sage: k = GF(7)
sage: k.characteristic()
7
Returns the degree of the finite field, which is a positive integer.
EXAMPLES:
sage: FiniteField(3).degree()
1
Return generator of this finite field as an extension of its prime field.
Note
If you want a primitive element for this finite field instead, use multiplicative_generator().
EXAMPLES:
sage: k = GF(13)
sage: k.gen()
1
sage: k.gen(1)
Traceback (most recent call last):
...
IndexError: only one generator
Return True since this is a prime field.
EXAMPLES:
sage: k.<a> = GF(3)
sage: k.is_prime_field()
True
sage: k.<a> = GF(3^2)
sage: k.is_prime_field()
False
Return the minimal polynomial of self, which is always .
EXAMPLES:
sage: k = GF(199)
sage: k.modulus()
x + 198
Return the order of this finite field.
EXAMPLES:
sage: k = GF(5)
sage: k.order()
5
Returns the polynomial name.
EXAMPLES:
sage: k.<a> = GF(3)
sage: k.polynomial()
x