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--- status: TODO
--- author(s):
--- notes:
document {
Key => {GF,[GF, Variable],[GF, PrimitiveElement],[GF, SizeLimit],(GF,ZZ,ZZ),(GF,ZZ),(GF,Ring),[GF,Strategy]},
Headline => "make a finite field",
SYNOPSIS (
BaseFunction => GF,
Usage => "GF(p,n)\nGF(q)",
Inputs => {
"p" => "a prime number", "n",
Variable => Symbol => {
"the name to use for the generator of the field. If null, ",
"then ", VAR "a", " is used."},
SizeLimit => ZZ => {
"the limit on the size of a Galois field whose elements will be represented
internally as powers of the primitive element"
}
},
Outputs => {
GaloisField => {"a finite field with ", TT "q = p^n", " elements"},
},
"The generator of this ring is a primitive element: it generates
the multiplicative group of non-zero elements.",
PARA{"If the single argument form GF(q) is given, q should be a prime power q = p^n"},
EXAMPLE lines ///
A = GF(3,2,Variable=>b);
ambient A
b^8
b^4
///,
EXAMPLE lines ///
K = GF 8
a^3+a
///
),
SeeAlso => {toField, "finite fields"},
SYNOPSIS (
BaseFunction => GF,
Usage => "GF R",
Inputs => {
"R" => Ring => {
"A quotient of a polynomial ring over ", TT "ZZ/p", " in one variable, modulo
an irreducible polynomial"
},
PrimitiveElement => {
"either an element of ", TT "R", ", or the symbol ", TO FindOne,
". An element is primitive if it generates the multiplicative group
of non-zero elements of R"
},
SizeLimit => ZZ => {
"the limit on the size of a Galois field whose elements will be represented
internally as powers of the primitive element"
}
},
Outputs => {
GaloisField => {"a finite field isomorphic to ", TT "R"}
},
EXAMPLE lines ///
A = ZZ/5[a]/(a^3-a-2)
B = GF A
C = ZZ/5[b]/(b^3+1+3*b^2+b)
D = GF C
map(B,D,{a^2})
///
),
}
document {
Key => isPrimitive,
Headline => "whether an element is a primitive element of a finite field",
TT "isPrimitive(f)", " -- Given an element ", TT "f", " in a quotient of a polynomial ring ",
TT "R", " over a finite field ", TT "K", "which is itself a finite field,
with the ring being finite dimensional over the field,
determine if ", TT "f", " generates the multiplicative group of this field.",
EXAMPLE { "R = ZZ/5[t]/(t^2+t+1);", "isPrimitive t", "isPrimitive (t-1)" }
}
document {
Key => order,
Headline => "a key used internally",
TT "order", " -- used as a key inside finite fields under which is
stored the number of elements in the field. Intended for internal use only",
PARA{},
SeeAlso => "GaloisField"
}
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