No, a degree in aerospace is not required, it just needs to be a bachelors degree from an ACCREDITED school. I emphasize that because if it is not accredited its value is worthless. As for getting to the majors, get your flight training done, build your time to 1500 hours, get your Airline Transport Pilot License, fly for the regionals and build that jet time and chase the jobs at the majors. It wont happen overnight but with the right persistence you will make it.