# 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_{p in P} p^{nu_p}, m = prod_{p in P} p^{mu_p} with mu_p leq nu_p, and { e_{n_1}^j}_{j in J} otimes { e_{n_2}^k}_{k in K} = { e_{n_1}^j cdot e_{n_2}^k}_{j in J, k in K}.

Then

[ calB_n,m = bigotimes_pin P bigotimes_k=mu_p^nu_p-1 { e_p^k+1^j}_jin J_k,p mboxrm where J_k,p = left{ beginarraylcl { 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 endarrayright. . ]

{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 ]```

GAP 3.4.4
April 1997