However, at most companies, there's no formal requirement that you do so. Most employment in the United States is called "at will", which means that your current employer can fire you at any time and for any reason (except those that are prohibited under law).
If You Want to Do More Than Just the Basics with Your Resignation Letter. You can always write more, and craft a longer resignation letter, but the truth is that it isn't necessary. You can add a portion of the note where you thank your manager or boss for the opportunities and time you have had together, or add that you hope to keep in touch, but in reality, if you really feel this way you probably should make an effort and find an opportunity to say these things in person (rather than getting an email).