The Nonexistence of Certain Galois Extensions Unramified Outside 5 Original Research Article

Letρbe a two-dimensional semisimple odd representation ofGal(image/image) over a finite field of characteristic 5 which is unramified outside 5. Assuming the GRH, we show in accordance with a conjecture by Serre thatρ=χa5circled plusχb5, wherea+bis odd.