implement cyclotomic norm residues

[was]

Here's the code, basically:

bug day 6 -- #1342 system:sage

K.<zeta> = CyclotomicField(7)

def norm_symbol_prime(a, P):
     K = P.number_field()
     zeta = K.gen()
     n = K.zeta_order()
     exponent = (1/n) * ( P.norm() - 1)
     exponent = ZZ(exponent)
     FF = K.residue_field(P)
     aa = FF(a)
     b = FF(a)^exponent
     zeta_mod = FF(zeta)
     # Find power m of zeta_mod that is equal to b, then
     # return zeta^m
     m = 0
     w = FF(1)
     while w != b:
         w = w * zeta_mod
         m += 1
         assert m <= n, "bug in norm_symbol_prime"
     return zeta^m

def norm_symbol(a, b):
     F = K.fractional_ideal([b]).factor()
     return prod([norm_symbol_prime(a, P)^e for P, e in F],
               K(1))

norm_symbol(zeta^3, 13*zeta)
///
-zeta^5 - zeta^4 - zeta^3 - zeta^2 - zeta - 1

norm_symbol(zeta^7, K(11))
///
1

norm_symbol((1+zeta)^2, 23*zeta)
///
zeta^4

[jdemeyer]

Please explain what are "cyclotomic norm residues".