# Gauss sums are special cases of Lagrange resolvents
#
# Gauss sums G(c)=Sum c(t) exp(2pi i t /p)
# 
p=7 # prime p
G=DirichletGroup(p); G # Group of Dirichlet characters
c=G[1]; c # select a generator
GS=c.gauss_sum(); GS # gauss sum of c(t)

# Since c(t) values are in Q(zeta(p-1)) and Fourier additive characters are in Q(zeta(p))
# the Gauss sum belongs to Q(zeta(p*(p-1)))
# If p=7 => p(p-1)=42
# 
# If the order of the character is m then the Gauss sum belongs to Q(zeta(p*m))
m=3 # a divisor of p-1
c1=c^ZZ((p-1)/m); c1
print "Order:", c1.order()
gs=c1.gauss_sum(); gs

# well ... check this is an element of Q(zeta(p*3))
# *************************************************************************

# Plotting Gauss sums as 0-complexes? i.e. formal sums of complex numbers
a = plot([],figsize=(3,3),title='Gauss Sums',frame=True,axes_labels=['$x$-axis','$y$-axis'])
a+=list_plot(gs,color='blue', size=20)
show(a)

# Why twelve points when the Gauss sum has 11 points? (sums of abs. value of coefficients)

# Other examples of Gauss sums ...
p=3; m=2; dv=divisors(p-1) # prime & order | p-1
print "p=", p, " divisors of p-1:", dv, " length:", len(dv) # prime p
G=DirichletGroup(p); G # Group of Dirichlet characters
c=G[1]; c # select a generator
c1=c^ZZ((p-1)/m); c1
print "Order of c(t):", c1.order()
gs=c1.gauss_sum(); print "Gauss sum g(c):", gs
a = plot([],figsize=(3,3),title='Gauss Sums',frame=True,axes_labels=['$x$-axis','$y$-axis'])
a+=list_plot(gs,color='blue', size=20)
show(a)