Cyclotomic function fields over finite fields with irreducible quadratic modulus

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>∗</m:mo> </m:msubsup> <m:mo>.</m:mo> </m:math> Fq²^*. 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.