"I have five history books."
Translation:У меня есть пять книг по истории.
From all the sentences in Duolingo and all the various comments about how and when to leave out есть, my understanding is that if there's some kind of qualifier (like an adjective or a number) beside the noun, you may omit есть. Because of that qualifier, it's like you have already acknowledged the existence or possession of that noun. So if I said : У него красный дом, I am already implying that I know or assume that he already has A house, I just don't know the details. My other impression is that if есть is put in when there is a qualifier, then it makes the sentence into an emphatic one. Eg. He does indeed have a red house!
I have noticed that my original formulation is a bit muddled. So is it just plain wrong to use книга о + prepositional? How do you then differentiate between a book pertaining to a specific field of study (e.g. a history book) and a book on a particular argument or problem within that field (e.g. a book about the influence of the Decembrist uprising on the Russian revolution)?
В русском языке существует определенная сочетаемость слов. По предмету, например, по истории, по математике, по физике. Либо о событии, о влиянии, о периоде. Book on a specific field of study(History, Math, Physics) would be used with по. A book on a particular argument or problem would be used with о/об.