text: why is this fact about linear fractional transformations true? "the automorphisms of the rational function field k(t) which fix k are precisely the linear fractional transformations determined by specifying t --> (at+b)/(ct+d)  for a,b,c,d in k and ad-bc not equal to 0."  why can't anything else be an automorphism that fixes k? automorphisms need to be invertible. if you had t go to any other rational function, say f(t), then f(t) must be invertible under compositions via another rational function. but there will be multiple solutions to z=f(t) unless f(t) is a fractional linear transformation.