On the automorphism group of Fermat fields (Q1912270)

This paper was written in the early seventies and circulated among interested mathematicians. In the nineties the author gave his permission for publication. The manuscript has been read and arranged (where necessary) by H. Stichtenoth. The Fermat field of exponent \(n\) over the field \(\Omega\) is the field \(\Omega (x, y)\) with \(x^n+ y^n+ 1=0\). There are some obvious automorphisms of \(\Omega (x, y)/ \Omega\), which generate a group \(G_n\) called the Fermat group. The main result of the paper is the following theorem: Let \(n\geq 4\) and let \(\Omega\) be algebraically closed of any characteristic \(p\). If \(n-1\) is not a power of \(p\), then \(G_n\) is the full automorphism group of \(K_n/ \Omega\). If \(n- 1 =q\) is a power of \(p\) then \(PGU (3, q^2)\) is the full automorphism group of \(\Omega (x, y)/ \Omega\).