Sure, you could omit credit — but why?
One of the simplest, most generous things to do in the world is give people credit. Ethically it’s also the right thing to do. And it’s not like it costs you any money or more than a minute of your time to put a line of credit in your book. So why not?
Many top-name authors *do* hire uncredited ghostwriters (or they’re hired by the publisher). But this is for strategic or logistical reasons — the author is too busy, the author’s time is too valuable, the author’s writing skills aren’t great, or the publisher wants more control over the process. Even in these situations, however, the ghostwriter is sometimes credited in an oblique way in the acknowledgments, such as “Thanks to Jane Schmidt for her great contributions in bringing this book to life.”
Note: Despite what I’m advocating, it’s always best to get the permission of the person you want to credit. For various reasons you may not know, she or he may want to remain anonymous or uncredited, or may want specific credit. For example, an author one time wanted to acknowledge me as his publisher, which I was not. It was a misunderstanding of terminology on his part, but it was an important distinction.
So, yes, you are not *required* to give credit if you’re paying for this person to do a service for you (and contract them as “work for hire”). You retain all rights to the content and can do with it as you wish.
But think whether it might be the right thing to acknowledge him or her.
Ethics aside, it’s always good to have yet another person out there willing to talk about your book. Credit your writer, and this is more likely to happen than if you don’t.