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For the full range of operations for elements of a number field or order
see Section Element Operations.
Subsections
Because of the nature of cyclotomic fields and orders, some properties of
elements are easier to determine than in the general case.
IsReal(a) : RngCycElt -> BoolElt
Whether the cyclotomic field or ring element a is a real number, i.e.,
if it is invariant under the complex conjugation.
Elements of cyclotomic fields and orders can additionally have their
complex conjugate computed. Conjugates are returned as cyclotomic elements
(and not reals) and which conjugate is wanted can be indicated by
providing a primitive root of unity.
ComplexConjugate(a) : RngCycElt -> RngCycElt
The complex conjugate of cyclotomic field or ring element a.
Conjugate(a, n) : RngCycElt, RngIntElt -> RngCycElt
The nth conjugate of a where 1 ≤n ≤ Degree(Parent(a)).
This is usually not the map ζ|-> ζn.
Conjugate(a, r) : RngCycElt, RngCycElt -> RngCycElt
The conjugate of the element a∈Q(ζm) or its order,
obtained by applying the
field automorphism ζm |-> r where r is a primitive root of unity.
The following lines of code generate a set W of minimal polynomials
for the so-called Gaussian periods
ηd=∑i=0(l - 1)/d - 1ζl^(g^((d)i))
where ζl is a primitive l-th root of unity (l is prime),
and where g is a primitive root modulo l. These
have the property that they generate a degree d cyclic subfield of
Q(ζl). We (arbitrarily) choose l=13 in this example.
> R<x> := PolynomialRing(RationalField());
> W := { R | };
> l := 13;
> L<z> := CyclotomicField(l);
> M := Divisors(l-1);
> g := PrimitiveRoot(l);
> for m in M do
> d := (l-1) div m;
> g_d := g^d;
> w := &+[z^g_d^i : i in [0..m-1] ];
> Include(~W, MinimalPolynomial(w));
> end for;
Here is the same loop in just one line, using sequence reduction:
> W := { R | MinimalPolynomial(&+[z^(g^((l-1) div m))^i : i in [0..m-1] ]) : > m in M };
> W;
{
x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 +
x + 1,
x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1,
x^3 + x^2 - 4*x + 1,
x + 1
x^4 + x^3 + 2*x^2 - 4*x + 3,
x^2 + x - 3,
}
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