EB( N ), EC( N ), ldots, EH( N ),
EI( N ), ER( N ),
EJ( N ), EK( N ), EL( N ), EM( N ),
EJ( N, d ), EK( N, d ), EL( N, d ), EM( N, d ),
ES( N ), ET( N ), ldots, EY( N ),
ES( N, d ), ET( N, d ), ldots, EY( N, d ),
NK( N, k, d )
For N a positive integer, let z = 'E(<N>)' = e^{2 pi i / N}. The following so-called atomic irrationalities (see~cite[Chapter 7, Section 10]CCN85) can be entered by functions (Note that the values are not necessary irrational.):
[beginarrayllllll
EB(N) & = & b_N & = & frac12sum_j=1^N-1z^j^2 &
(Nequiv 1bmod 2)
EC(N) & = & c_N & = & frac13sum_j=1^N-1z^j^3 &
(Nequiv 1bmod 3)
ED(N) & = & d_N & = & frac14sum_j=1^N-1z^j^4 &
(Nequiv 1bmod 4)
EE(N) & = & e_N & = & frac15sum_j=1^N-1z^j^5 &
(Nequiv 1bmod 5)
EF(N) & = & f_N & = & frac16sum_j=1^N-1z^j^6 &
(Nequiv 1bmod 6)
EG(N) & = & g_N & = & frac17sum_j=1^N-1z^j^7 &
(Nequiv 1bmod 7)
EH(N) & = & h_N & = & frac18sum_j=1^N-1z^j^8 &
(Nequiv 1bmod 8)
(Note that in c_N, ldots, h_N, N must be a prime.)
[beginarraylllll
ER(N) & = & sqrtN
EI(N) & = & i sqrtN & = & sqrt-N
From a theorem of Gauss we know that
[ b_N = left{ beginarrayllll
frac12(-1+sqrtN) & rm if & Nequiv 1 & bmod 4
frac12(-1+isqrtN) & rm if & Nequiv -1 & bmod 4
endarrayright. ,]
so sqrt{N} can be (and in fact is) computed from b_N. If N is a
negative integer then ER(N) = EI(-N).
For given N, let n_k = n_k(N) be the first integer with multiplicative order exactly k modulo N, chosen in the order of preference [ 1, -1, 2, -2, 3, -3, 4, -4, ldots .]
We have
[beginarrayllllll
EY(N) & = & y_n & = & z+z^n &(n = n_2)
EX(N) & = & x_n & = & z+z^n+z^n^2 &(n=n_3)
EW(N) & = & w_n & = & z+z^n+z^n^2+z^n^3 &(n=n_4)
EV(N) & = & v_n & = & z+z^n+z^n^2+z^n^3+z^n^4 &(n=n_5)
EU(N) & = & u_n & = & z+z^n+z^n^2+ldots +z^n^5 &(n=n_6)
ET(N) & = & t_n & = & z+z^n+z^n^2+ldots +z^n^6 &(n=n_7)
ES(N) & = & s_n & = & z+z^n+z^n^2+ldots +z^n^7 &(n=n_8)
[beginarrayllllll
EM(N) & = & m_n & = & z-z^n &(n=n_2)
EL(N) & = & l_n & = & z-z^n+z^n^2-z^n^3 &(n=n_4)
EK(N) & = & k_n & = & z-z^n+ldots -z^n^5 &(n=n_6)
EJ(N) & = & j_n & = & z-z^n+ldots -z^n^7 &(n=n_8)
Let n_k^{(d)} = n_k^{(d)}(N) be the d+1-th integer with
multiplicative order exactly k modulo N, chosen in the order of
preference defined above; we write
n_k=n_k^{(0)},n_k^{prime}=n_k^{(1)}, n_k^{primeprime} = n_k^{(2)}
and so on. These values can be computed as NK(N,k,d) =
n_k^{(d)}(N); if there is no integer with the required multiplicative
order, NK will return false.
The algebraic numbers [y_N^prime=y_N^(1),y_N^primeprime=y_N^(2),ldots, x_N^prime,x_N^primeprime,ldots, j_N^prime,j_N^primeprime,ldots] are obtained on replacing n_k in the above definitions by n_k^{prime},n_k^{primeprime},ldots; they can be entered as
[beginarraylll
EY(N,d) & = & y_N^(d)
EX(N,d) & = & x_N^(d)
& vdots
EJ(N,d) & = & j_n^(d)
gap> EW(16,3); EW(17,2); ER(3); EI(3); EY(5); EB(9);
0
E(17)+E(17)^4+E(17)^13+E(17)^16
-E(12)^7+E(12)^11
E(3)-E(3)^2
E(5)+E(5)^4
1
GAP 3.4.4