S4andS4Extensions of image Ramified at Only One Prime Original Research Article

For each primepin a certain family of odd primes, we construct anS4extension of image unramified outsidep. We show that for allp≡3 (mod 8) in our family, thisS4extension embeds in anS4extension, which is also unramified outsidep. Invoking Serreʹs conjecture (in a proven case) allows us to relate the splitting of primes in these extensions to certain modular forms of level 1.