We consider the operation of the symmetric group S4 onthe set R[x,y,z,a] through permutation of an unknown integer.
a) Calculate the length of the orbit of polynomial x2+y2+z+a. Howmany permutations leave this polynomial unchanged?
b) Is a polynomial of length 5 under this operation possible?
c) Show the existence of polynomials with orbit length 12 and4.
