13.1 More about Cyclotomics

Subfields of Cyclotomic Fields), cyclotomics for short, are arithmetical objects like rationals and finite field elements; they are not implemented as records ---like groups--- or e.g. with respect to a character table (although character tables may be the main interest for cyclotomic arithmetics).

E( n )

returns the primitive n-th root of unity e_n = e^{frac{2pi i}{n}}. Cyclotomics are usually entered as (and irrational cyclotomics are always displayed as) sums of roots of unity with ATLAS irrationalities.)

    gap> E(9); E(9)^3; E(6); E(12) / 3; 
    -E(9)^4-E(9)^7    # the root needs not to be an element of the base
    E(3)
    -E(3)^2
    -1/3*E(12)^7

For the representation of cyclotomics one has to recall that the cyclotomic field Q_n = Q(e_n) is a vector space of dimension varphi(n) over the rationals where varphi denotes Euler's phi-function (see Phi).

Note that the set of all n-th roots of unity is linearly dependent for n > 1, so multiplication is not the multiplication of the group ring Q < e_n > ; given a Q-basis of Q_n the result of the multiplication (computed as multiplication of polynomials in e_n, using (e_n)^n = 1) will be converted to the base.

    gap> E(5) * E(5)^2; ( E(5) + E(5)^4 ) * E(5)^2;
    E(5)^3
    E(5)+E(5)^3
    gap> ( E(5) + E(5)^4 ) * E(5);
    -E(5)-E(5)^3-E(5)^4

Cyclotomics are always represented in the smallest cyclotomic field they are contained in. Together with the choice of a fixed base this means that two cyclotomics are equal if and only if they are equally represented.

Addition and multiplication of two cyclotomics represented in Q_n and Q_m, respectively, is computed in the smallest cyclotomic field containing both: Q_{'Lcm'(n,m)}. Conversely, if the result is contained in a smaller cyclotomic field the representation is reduced to the minimal such field.

The base, the base conversion and the reduction to the minimal cyclotomic field are described in~Zum89, more about the base can be found in ZumbroichBase.

Since n must be a short integer, the maximal cyclotomic field implemented in GAP is not really the field Q^{ab}. The biggest allowed (though not very useful) n is 65535.

There is a global variable Cyclotomics in GAP, a record that Subfields of Cyclotomic Fields).

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