Element Operations

For the full range of operations for elements of a number field or order see Section Element Operations.

Predicates on Elements

Because of the nature of cyclotomic fields and orders, some properties of elements are easier to determine than in the general case.

IsReal(a): FldCycElt -> BoolElt

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.

Conjugates

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): FldCycElt -> FldCycElt

ComplexConjugate(a): RngCycElt -> RngCycElt

The complex conjugate of cyclotomic field or ring element \(a\).

Conjugate(a, n): FldCycElt, RngIntElt -> FldCycElt

Conjugate(a, n): RngCycElt, RngIntElt -> RngCycElt

The image under the map \(\zeta \mapsto \zeta^n\). The second argument (\(n\)) must be coprime to the conductor.

Conjugate(a, r): FldCycElt, FldCycElt -> FldCycElt

Conjugate(a, r): RngCycElt, RngCycElt -> RngCycElt

The conjugate of the element \(a\in{\mathbb{Q}}(\zeta_m)\) or its order, obtained by applying the field automorphism \(\zeta_m\mapsto r\) where \(r\) is a primitive root of unity.

Example: Gaussian Periods (ex-591c1c)

The following lines of code generate a set \(W\) of minimal polynomials for the so-called Gaussian periods

\[\eta_d=\sum_{i=0}^{(l-1)/d-1}\zeta_l^{g^{{d}i}}\]

where \(\zeta_l\) 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 \({\mathbb{Q}}(\zeta_l)\). 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;

Run in calculatorCopy input

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,
}

Run in calculatorCopy input