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#                   Gauss Sums 
#
# Theory:     g_m(c)=Sum c(x) zeta(mx); m=1 => g_m(c)=c(m^(-1)) g(c)
#              where c:Fp^x -> Q(zeta(p-)
# SAGE:       g(c): c.gauss_sum()
#####################################################################
G = DirichletGroup(13); G
Group of Dirichlet characters of modulus 13 over Cyclotomic Field of order 12 and degree 4 
# Pick a Dirichlet character
e = G.0; e # This Dirichlet character is a generator, bcause x=2 generates Fp*
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12 
# Computes the Gauss sums
g=e.gauss_sum(); g # It is the basic Gauss sum corresponding to m=1 above
-zeta156^46 + zeta156^45 + zeta156^42 + zeta156^41 + 2*zeta156^40 + zeta156^37 - zeta156^36 - zeta156^34 - zeta156^33 - zeta156^31 + 2*zeta156^30 + zeta156^28 - zeta156^24 - zeta156^22 + zeta156^21 + zeta156^20 - zeta156^19 + zeta156^18 - zeta156^16 - zeta156^15 - 2*zeta156^14 - zeta156^10 + zeta156^8 + zeta156^7 + zeta156^6 + zeta156^5 - zeta156^4 - zeta156^2 - 1 
# Note: why zeta156 !? [see below after Example 2]
# The norm of Gauss sums is square root of the prime modulus
#  |g|^2=p
factor(norm(g))
13^24 
# Example 2
DP=DirichletGroup(5); DP
Group of Dirichlet characters of modulus 5 over Cyclotomic Field of order 4 and degree 2 
e=DP.0; e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4 
g=e.gauss_sum(); g
zeta20^7 + zeta20^6 + zeta20^4 - zeta20^3 
# SAGE computes Gauss sums as elements of Fp[zeta(p-1)] 
# which is viewed as elements of Q(zeta(p*(p-1)))
# Example: p=13 => p(p-1)=156; p=5 => p(p-1)=20
# 

# Example 3
p=3; p
DP=DirichletGroup(3); DP
3 Group of Dirichlet characters of modulus 3 over Cyclotomic Field of order 2 and degree 1 
c=DP[0]; c
print "Gauss sum ", c.gauss_sum()
Dirichlet character modulo 3 of conductor 1 mapping 2 |--> 1 Gauss sum -1 
c=DP[1]; c
c.gauss_sum()
Dirichlet character modulo 3 of conductor 3 mapping 2 |--> -1 2*zeta6 - 1 
# Example 4
DP=DirichletGroup(7); c=DP[1]; c.gauss_sum()
zeta42^10 + zeta42^8 + zeta42^6 - zeta42^5 - zeta42^4 - zeta42^2 - zeta42