Genus zero translates of three point ramified Galois extensions (Q802670)

Let k be a number field, k(t) a simple transcendental extension and N a regular Galois extension of k(t) ramified at three points. If L is an extension of k(t), LN/L is called translated field extension. In this paper when L is special the author derives sufficient condition for LN/L to be defined over a proper subfield of k and uses these to get some Galois extensions of \({\mathbb{Q}}(t)\). As a consequence for an infinite set of primes p, \(L_ 2(p)\) are realizable as Galois groups over \({\mathbb{Q}}\).