GAP Manual: 15.9. ZumbroichBase

ZumbroichBase( n, m )

returns the set of exponents i where e_n^i belongs to the base (cal(B))_(n,m) of the field extension Q_n/Q_m; for that, n and m must be positive integers where m divides n.

(cal(B))_(n,m) is defined as follows:

Let P denote the set of prime divisors of n, n = prod_(pinP) p^(nu_p), m = prod_(pinP) p^(mu_p) with mu_p leqnu_p, and { e_(n_1)^j}_(jinJ) otimes{ e_(n_2)^k}_(kinK) = { e_(n_1)^j cdote_(n_2)^k}_(jinJ, kinK).

Then

[ calB_n,m = bigotimes_pinP bigotimes_k=mu_p^nu_p-1 { e_p^k+1^j}_jinJ_k,p mboxrm where J_k,p = left{ lcl { 0 } & ; & k=0, p=2
{ 0, 1 } & ; & k > 0, p=2
{ 1, ldots, p-1 } & ; & k = 0, pnot= 2
{ -fracp-12, ldots, fracp-12 } & ; & k > 0, pnot= 2 right. . ]

(cal(B))_(n,1) is equal to the base (cal(B))(Q_n) of Q_n over the rationals given in Zum89 (Note that the notation here is slightly different from that there.).

(cal(B))_(n,m) consists of roots of unity, it is an integral base (that is, the integral elements in Q_n have integral coefficients, see Cyclotomic Integers), it is a normal base for squarefree n and closed under complex conjugation for odd n. 

    gap> ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );
    [ 1, 2, 4, 7, 8, 11, 13, 14 ]
    [ 0, 3 ]
    gap> ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );
    [ 2, 4, 6, 8 ]
    [ 0, 1, 2, 3, 4, 5, 6, 7 ]
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