GAP Manual: 13.12. ATLAS irrationalities

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 pii / N). The following so-called atomic irrationalities (see Chapter 7, Section 10]{CCN85) can be entered by functions (Note that the values are not necessary irrational.):

[llllll EB(N) & = & b_N & = & frac12sum_j=1^N-1z^j^2 & (Nequiv1bmod2)

EC(N) & = & c_N & = & frac13sum_j=1^N-1z^j^3 & (Nequiv1bmod3)

ED(N) & = & d_N & = & frac14sum_j=1^N-1z^j^4 & (Nequiv1bmod4)

EE(N) & = & e_N & = & frac15sum_j=1^N-1z^j^5 & (Nequiv1bmod5)

EF(N) & = & f_N & = & frac16sum_j=1^N-1z^j^6 & (Nequiv1bmod6)

EG(N) & = & g_N & = & frac17sum_j=1^N-1z^j^7 & (Nequiv1bmod7)

EH(N) & = & h_N & = & frac18sum_j=1^N-1z^j^8 & (Nequiv1bmod8) ]

(Note that in c_N, ldots, h_N, N must be a prime.)

[lllll ER(N) & = & sqrtN
EI(N) & = & i sqrtN & = & sqrt-N
]

From a theorem of Gauss we know that [ b_N = left{ llll frac12(-1+sqrtN) & rmif & Nequiv1 & bmod4
frac12(-1+isqrtN) & rmif & Nequiv-1 & bmod4 right. ,]

so sqrt(N) can be (and in fact is) computed from b_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 [llllll 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) ]

[llllll 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

[lll 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); ER(-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
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