No.  Legally, you establish ownership of your work (i.e. secure copyright) the moment you fix it into tangible form—e.g., in a doc. file, onto paper, a recording, and so on. 

No publication or registration is required. However, there are sometimes good reasons to register copyright with the Library of Congress, especially if your work might have commercial value, or if you are just concerned that someone will steal your stuff.  You can register the work with the Library of Congress easily enough.  See my favorite document on the subject of all things copyright: https://www.copyright.gov/circs/circ01.pdf.

One case:  I know of one notorious case of a prominent poet whose poems were copied whole and republished by a fellow using a pseudonym, which did result in emotional trauma for the authentic poet, all of this documented in a book later.  The case was notorious, however, because of its extreme rarity; the fact that the work was copyrighted in book form did nothing to stop the plagiarism.

One tip:  Writers who put a copyright notice at the top of their unpublished manuscripts, when submitting work to a magazine or press, tip off the editor that he or she is dealing with a true amateur.  Don’t do it.  It isn’t useful and you will look like someone who doesn’t know the business.

- Robert Stewart