Can you publish a book written about something like the American Idol?

From what I've seen, most people when referencing something in the real world alter the name. Shows from Law and Order to iCarly have done it and I'm sure the same applies to books. To avoid any copy write issues just change the name from American Idol to Singing Idol or something like that. So long as the characteristics of the show remain the same people will know what you're talking about.