According to my attorney wife (who does a bit in intellectual property), the US provides some protection for original work even if copyright is not overtly stated (e.g., using the circled "c" mark or stating that the material is copyrighted). Assertion of copyright provides more protection, and no, you do not have to file with the Copyright Office at the Library of Congress to have some copyright protection.

Not asserting copyright can make it harder to prove your case for being the original author in court if it should come to that, however.