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Element Operations






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




Subsections



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 nth conjugate of a where 1 <= n <= Degree(Parent(a)).
This is usually not the map zeta |-> zeta^n.



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


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



The conjugate of the element a in Q(zeta_m) or its order, 
obtained by applying the 
field automorphism zeta_m |-> r where r is a primitive root of unity.





Example FldCyc_GaussianPeriods (H53E2)

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
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;


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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