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# Multiplicative Characters of a finite field Fp:
# c:Fp*->C*, c(ab)=c*(a)c(b)
#
# prime number p=7
G=DirichletGroup(7); G # Finite field Fp
G.gen(0).base_ring() # R is a cyclotomic field; see SAGE number fields
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2 Cyclotomic Field of order 6 and degree 2 
# ... this is Q(z6), and the degree of the extension [Q(z6):Q] is 2  (never mind this ... for now)

n=len(G); n # the LENGTH of the list is the number of characters: same as |Fp*|=p-1
for k in range(n):
    print(G[k])
6 Dirichlet character modulo 7 of conductor 1 mapping 3 |--> 1 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -1 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -zeta6 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -zeta6 + 1 
# Write by hand the correspondence, compute some values and represent graphically ...
#

# *************************************************************************
#             Generators of the group of Dirichlet characters
# *************************************************************************
x=G.gens(); x # independent generators: ONE per dimension!!
d=len(x); d # number of independent generators
for k in range(d):
    x[k]
(Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6,) 1 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6