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Dom::GaloisField(p, n, f)(
Dom::GaloisField(p, n, f) creates the residue
class field ,
a finite field with pn elements.
f is not given, it is chosen at random among
all irreducible polynomials of degree n.
Dom::GaloisField(q) (where q = pn)
is equivalent to
Dom::GaloisField(F, n, f) creates the residue
class field F[X]/<f>,
a finite field with |F|n elements.
f is not given, a random irreducible polynomial
of appropriate degree is used; some free identifier is chosen as its
variable, and this one must also be used when creating domain elements.
Although n = 1 is
be used for representing prime fields.
F is of type
consisting of residue classes of polynomials, the variable of these
polynomials must be distinct from the variable of
If a tower several of Galois fields is constructed, the variable used
in the uppermost Galois field must not equal any of those used in
the tower. A special entry
to keep track of all variables appearing somewhere in the tower.
Dom::GaloisField(p,n,f)(g) (or, respectively,
creates the residue class of
It is represented by the unique polynomial in that class that has
smaller degree than
L to be the field with 4 elements.
Then a4 = a for
every a ∈ L,
by a well-known theorem.
L:=Dom::GaloisField(2, 2, u^2+u+1): L(u+1)^4
Univariate irreducible polynomial over
Finite field of type
Univariate polynomial over the ground field in the same variable
the zero element of the field
the unit element of the field
the characteristic of the field
the number of elements of the field
the prime field, which equals
the variable of the polynomial
a list consisting of
iszero— Test for zero
It overloads the function
_power— Integer power of an element
frobenius— Frobeniu map
conjugates— Conjugate of an element
order— Order of an element
isSquare— Test whether an element is a square
ln— Discrete logarithm
elementNumber— Enumerate field elements
The inverse of this mapping has not been implemented.
companionMatrix— Companion matrix of the Galois field over its ground field
companionPowers— Power of the companion matrix
matrixRepresentation— Isomorphism to the algebra generated by the companion matrix
If A is the companion matrix, the image of is .
randomPrimitive— Choose a primitive element at random
isBasis— Test elements for being a basis over the ground field
isNormal— Test whether a given field element is normal
randomNormal— Choose normal element at random
isPrimitivePolynomial— Test whether a polynomial over the field is primitive
convert— Conversion from other types
convert_to— Conversion to other types