Can you be arrested for saying that you have killed someone? [US/UK]

Lets say hypothetically that you tell law enforcement that you've killed someone, no one in particular, just that you have murdered someone, and you describe in detail what you did, leaving out names and specifics etc... Can they arrest you for this? Without having any evidence? Is your testimony enough?

Further extending this, if you tell someone that you've committed a crime, but leave out specifics, can you get arrested for this?

Thanks.