Finite fields
Dom::GaloisField(q
)
Dom::GaloisField(p
,n
)
Dom::GaloisField(p
,n
,f
)
Dom::GaloisField(F
,n
)
Dom::GaloisField(F
,n
,f
)
Dom::GaloisField(p, n, f)(g
)
Dom::GaloisField(p, n, f)
creates the residue
class field
,
a finite field with p^{n} elements.
If f
is not given, it is chosen at random among
all irreducible polynomials of degree n.
Dom::GaloisField(q)
(where q = p^{n})
is equivalent to Dom::GaloisField(p,n)
.
Dom::GaloisField(F, n, f)
creates the residue
class field F[X]/<f>,
a finite field with F^{n} elements.
If 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
allowed, Dom::IntegerMod
should
be used for representing prime fields.
If F
is of type Dom::GaloisField
,
consisting of residue classes of polynomials, the variable of these
polynomials must be distinct from the variable of f
.
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 "VariablesInUse"
serves
to keep track of all variables appearing somewhere in the tower.
Dom::GaloisField(p,n,f)(g)
(or, respectively, Dom::GaloisField(F,n,f)(g)
)
creates the residue class of g
modulo f
.
It is represented by the unique polynomial in that class that has
smaller degree than f
.
Cat::Field
, Cat::Algebra
(F)
, Cat::VectorSpace
(F)
We define L
to be the field with 4 elements.
Then a^{4} = a for
every a ∈ L,
by a wellknown theorem.
L:=Dom::GaloisField(2, 2, u^2+u+1): L(u+1)^4

Prime power 

Prime 

Positive integer 

Univariate irreducible polynomial over 

Finite field of type 

Univariate polynomial over the ground field in the same variable
as 
"zero"  the zero element of the field 
"one"  the unit element of the field 
"characteristic"  the characteristic of the field 
"size"  the number of elements of the field 
"PrimeField"  the prime field, which equals 
"Variable"  the variable of the polynomial 
"VariablesInUse"  a list consisting of 