This semester I have the course "Algebra III" in university. Our textbook is "Algebra Volume I: Fields and Galois Theory by Falko Lorenz. It's a very useful book but the problem is that it doesn't have solution manual, so I don't have any reference to check whether my answers to its questions are true or not. However; I'm looking for another textbook in which I can find all Lorenz's book subjects and topics and also its solution manual is available.
Any help and idea would be appreciated.
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Apart from Milne's notes, I don't think there is any book,let alone a GOOD book in Galois Theory with solution manual.
There are plenty of good textbooks on Galois Theory and already a lot of threads concerning that.However, I think most Galois Theory books don't motivate the importance Galois Theory plays in modern mathematics. An excellent book to read about another face of Galois Theory might be a book called The Galois Dream. It is also wise to learn some algebraic number theory along the way as you will se many of the ideas you develop in your course applied there.
If your are more advanced, you can start looking at Galois Groups and Fundamental Groups just for fun, they give a quick survey in Galois Theory in the first chapter, and go on with harder stuff later.
On a final note. It's better if you can discuss your solutions with a professor or some classmates than just looking at their solution to check if what you've done is correct.
You could try reading "Fields and Galois theory" by Patrick Morandi. You can also read the book with the same name by John.M.Howie.
Edit I however personally recommend the course notes of J.S.Milne. I have it here in this link. He explains it very nicely and there are solutions as well.