#
###################################################################
# Chapter 6, Exercise 7, page 274, GCL.
#
# CORRECTION: The sign in front of the term x*y^2*z should be minus (-)
# rather than plus (+) as printed in the textbook. Also, a(x) is actually
# a(x,y,z).
#
#
# This is a hint for students -- not a complete solution here.
###################################################################
#
`mod` := mods;
p := 3;
#
#=================================================
# We want to solve  F(a) = a^2 - a - r = 0  where
#=================================================
#
r := x^6 + x^4*y^2 + x^3*z + x^2*y^4 - x*y^2*z + z^2 - 1;
#
#=================================================
# Reduce mod I.
#=================================================
#
Id := [y=1,z=0];  # This is our representation of the ideal I = <y-1,z> .
rmodI := eval(r, Id) mod p;
#
#=========================================================================
# How to determine a solution in  Z_3[x,y,z] / I  ==  Z_3[x] ?
#=========================================================================
#
# Well, note that over Z_3,  (a+1)^2 = a^2 - a + 1.
#
# Therefore since F (mod I) takes the form
#
FmodI := a^2 - a - rmodI;
#
# we can restate the problem  FmodI = 0  in the form
#
#    (a+1)^2  =  x^6 + x^4 + x^2
#             =  x^2 * (x^4 + x^2 + 1)
#             =  x^2 * (x^2 - 1)^2        in Z_3[x] .
#
# I.e.   a+1  =  x * (x^2 - 1)  =  x^3 - x .
#
# Hence the desired solution  a (mod I)  is:
#
amodI := x^3 - x - 1;
#
#=================================================
# Check this claimed solution.
#=================================================
#
eval(FmodI, a=amodI);
Expand(%) mod p;
#
#=========================================================================
# Lifting step -- Newton's iteration:  anew = a - F(a)/F'(a) .
#=========================================================================
#
#  ... try it "by hand"
#  ... use (6.34)-(6.35)
#