Yes, I've read that book once, it's so boring, I didn't even get to its middle.

[Tarheel]

"Have you already read the book today?" is a perfectly grammatical sentence. That is, it follows the rules of English grammar. However, if you said that to me I would have to ask you what you meant.

Abe: Have you already read the book today?
Bob: Yeah, well, I read one chapter.
Abe: OK.
Bob: What do you mean by "already"?
Abe: I mean "Have you read it yet?"
Bob: Yes, I have.

 

[BestBuddy]

"Have you already read the book today?" is a perfectly grammatical sentence. That is, it follows the rules of English grammar. However, if you said that to me I would have to ask you what you meant.

Abe: Have you already read the book today?
Bob: Yeah, well, I read one chapter.
Abe: OK.
Bob: What do you mean by "already"?
Abe: I mean "Have you read it yet?"
Bob: Yes, I have.

Thanks!
What's the difference between 'yet' and 'already'?
"Have you read your book yet?"
"Have you already read your book?"
(Saying about a session of reading that has to have place every day in order to complete the book in several days in the nearest future)

 

[Tarheel]

If I say "Have you done it already?" I am expressing surprise that you did it (whatever it is) so soon. (It's not really a question.) Another example might be that I'm waiting for a bus and it comes before I'm expecting it. Maybe I thought there was ten minutes to go before the bus was supposed to show up. Somebody tells me the bus has arrived. I might say, "Is it here already?"

The words "yet" and "already" do not mean the same thing.

www.dictionary.com

 

[jutfrank]

Just to be clear, BestBuddy, your question should be either Have you read your book yet? Or perhaps better: Have you read your book today?

Note that these two questions don't quite have the same effect: the question with today carries the sense that you expect your child to undertake this task every day, while the question with yet carries the idea that if he hasn't read his book, then you still expect him to do it before the day ends.