) |
gen,
is_field,
is_finite,
ngens,
order,
polynomial,
polynomial_ring,
random_element,
unit_group_exponent,
vector_space
Further documentation:
) |
Returns whether or not the finite field is a field, i.e., always returns True.
sage: k = FiniteField(3**4) sage: k.is_field() True
) |
The number of generators of the finite field. Always 1.
sage: k = FiniteField(3**4) sage: k.ngens() 1
) |
Returns the polynomial ring over the prime subfield in the same variable as this finite field.
sage: k = FiniteField(3**4, "alpha") sage: k.polynomial_ring() Univariate Polynomial Ring in alpha over Finite field of size 3
[bound=None]) |
A random element of the finite field.
INPUT: bound -- ignored
sage: k = GF(2**10, 'a') sage: k.random_element() a^9 + a
) |
The exponent of the unit group of the finite field. For a finite field, this is always the order minus 1.
sage: k = GF(2**10) sage: k.order() 1024 sage: k.unit_group_exponent() 1023
Instances of class FiniteField_generic also have the following special methods:
__cmp__,
__getitem__,
__iter__,
_latex_
Further documentation:
other) |
Compares this finite field with other. Two finite fields are equal if and only if they have the same cardinality *and* the defining polynomials are the same.
sage: FiniteField(3**2) == FiniteField(3**3) False sage: FiniteField(3**2) == FiniteField(3**2) True sage: FiniteField(3**2,'beta') == FiniteField(3**2,'alpha') False sage: FiniteField(3**2,'beta') == FiniteField(3**2,'beta') True
n) |
Returns
-th element of the field. There is no guarantee
regarding how the elements of the field are ordered.
sage: k = GF(8, 'a') sage: k[0] 0 sage: k[1] 1 sage: k[7] a^2 + a + 1
) |
Return iterator over elements of the finite field, ending with 0.
sage: k = GF(9, 'a') sage: i = 0 sage: for x in k: print x, k[i]; i+= 1 0 0 1 1 2 2 a a a + 1 a + 1 a + 2 a + 2 2*a 2*a 2*a + 1 2*a + 1 2*a + 2 2*a + 2
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