For the full range of operations for elements of a number field or order see Section Element Operations.
Because of the nature of cyclotomic fields and orders, some properties of elements are easier to determine than in the general case.
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.
The complex conjugate of cyclotomic field or ring element a.
The image under the map ζ |-> ζn. The second argument (n) must be coprime to the conductor.
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.
> 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,
}