Abstract Let 𝔽 q be the finite field of order q and F = 𝔽 q (x) the rational function field. In this paper, we give a characterization of the cyclotomic function fields F (Λ M) with modulus M, where M ∈ 𝔽 q T is a monic and irreducible polynomial of degree two. More precisely, we show that F (Λ M) is the only function field, up to 𝔽 q -isomorphism, with q + 1 𝔽 q -rational places, genus (q + 1) (q − 2) /2 and a subgroup of automorphisms over 𝔽 q isomorphic to F q 2 ∗. Fₐℂ^*. We also provide the full automorphism group of F (Λ M) in odd characteristic, extending results of 14 where the automorphism group of F (Λ M) over 𝔽 q was computed.
Arakelian et al. (Sat,) studied this question.