18.4 ATLAS Irrationalities

  • EB( n ) F
  • EC( n ) F
  • ED( n ) F
  • EE( n ) F
  • EF( n ) F
  • EG( n ) F
  • EH( n ) F

    For N a positive integer, let z = E(N ) = e2 pi / N. The following so-called atomic irrationalities (see Chapter 7, Section 10 of CCN85) can be entered using functions. (Note that the values are not necessary irrational.)

    EB(N )
    =
    bN
    =
    1
    2
    		N-1
    å
    j = 1 
    		zj2
    (N º 1 mod 2)
    EC(N )
    =
    cN
    =
    1
    3
    		N-1
    å
    j = 1 
    		zj3
    (N º 1 mod 3)
    ED(N )
    =
    dN
    =
    1
    4
    		N-1
    å
    j = 1 
    		zj4
    (N º 1 mod 4)
    EE(N )
    =
    eN
    =
    1
    5
    		N-1
    å
    j = 1 
    		zj5
    (N º 1 mod 5)
    EF(N )
    =
    fN
    =
    1
    6
    		N-1
    å
    j = 1 
    		zj6
    (N º 1 mod 6)
    EG(N )
    =
    gN
    =
    1
    7
    		N-1
    å
    j = 1 
    		zj7
    (N º 1 mod 7)
    EH(N )
    =
    hN
    =
    1
    8
    		N-1
    å
    j = 1 
    		zj8
    (N º 1 mod 8)

    (Note that in cN, ¼, hN, N must be a prime.)

  • EI( n ) F
  • ER( n ) F

    For a rational number N, ER returns the square root Ö{N } of N, and EI returns Ö{-N }. By the chosen embedding of cyclotomic fields into the complex numbers, ER returns the positive square root if N is positive, and if N is negative then ER(N) = EI(-N). In any case, EI(N) = E(4) * ER(N).

    ER is installed as method for the operation Sqrt (see Sqrt) for rational argument.

    From a theorem of Gauss we know that

    bN = ì
    ï
    ï
    í
    ï
    ï
    î
    1
    2
    		(-1+ÖN)
    if
    N º 1
    mod 4
    1
    2
    		(-1+iÖN)
    if
    N º -1
    mod 4
    		 ,
    so ÖN can be computed from bN (see EB).

  • EY( n[, d] ) F
  • EX( n[, d] ) F
  • EW( n[, d] ) F
  • EV( n[, d] ) F
  • EU( n[, d] ) F
  • ET( n[, d] ) F
  • ES( n[, d] ) F

    For given N, let nk = nk(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, ¼.

    We define

    EY(N )
    =
    yn
    =
    z+zn
    (n = n2)
    EX(N )
    =
    xn
    =
    z+zn+zn2
    (n = n3)
    EW(N )
    =
    wn
    =
    z+zn+zn2+zn3
    (n = n4)
    EV(N )
    =
    vn
    =
    z+zn+zn2+zn3+zn4
    (n = n5)
    EU(N )
    =
    un
    =
    z+zn+zn2+¼+zn5
    (n = n6)
    ET(N )
    =
    tn
    =
    z+zn+zn2+¼+zn6
    (n = n7)
    ES(N )
    =
    sn
    =
    z+zn+zn2+¼+zn7
    (n = n8)

  • EM( n[, d] ) F
  • EL( n[, d] ) F
  • EK( n[, d] ) F
  • EJ( n[, d] ) F

    EM(N )
    =
    mn
    =
    z-zn
    (n = n2)
    EL(N )
    =
    ln
    =
    z-zn+zn2-zn3
    (n = n4)
    EK(N )
    =
    kn
    =
    z-zn+¼-zn5
    (n = n6)
    EJ(N )
    =
    jn
    =
    z-zn+¼-zn7
    (n = n8)

  • NK( n, k, d ) F

    Let nk(d) = nk(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 nk = nk(0),nk¢ = nk(1), nk¢¢ = nk(2) and so on. These values can be computed as NK(N,k,d) = nk(d)(N); if there is no integer with the required multiplicative order, NK returns fail.

    The algebraic numbers

    yN¢ = yN(1),yN¢¢ = yN(2),¼, xN¢,xN¢¢,¼, jN¢,jN¢¢,¼
    are obtained on replacing nk in the above definitions by nk¢,nk¢¢,¼; they can be entered as

    EY(N ,d )
    =
    yN(d)
    EX(N ,d )
    =
    xN(d)
    :
    EJ(N ,d )
    =
    jn(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

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    GAP 4 manual
    February 2000