On automorphism groups of cyclotomic function fields over finite fields

The Weierstrass semigroup of the unique totally ramified place in the cyclotomic function field with modulus xn+1xn+1 over the rational function field Fq(x)Fq(x) is explicitly computed for each positive integer n  . As a consequence, the automorphism groups of cyclotomic function fields with modulus xn+1xn+1 over finite fields can be determined. Similarly, the automorphism groups of the cyclotomic function fields with modulus P where P is an irreducible quadratic polynomial over finite fields are investigated as well.