It is probably a brass core, and it is definitely cheaper to replace it than install a new faucet. But, whether YOU can do it depends on how tight it is stuck in the valve. I have six different tools to remove it, and sometimes it takes more than one to accomplish the task. IF you start hammering, chiseling, or drilling it, you WILL have to replace the entire valve. Call a good plumber who has the necessary tools to remove it. Once it is removed, there is absolutely no reason to reuse the old one.