I mentioned in lecture something called the Hasse Principle, which holds for simple kinds of equations. You can read about this in the book ‘A course in arithmetic’ by J.-P. Serre. I highly recommend this book as an introduction to the theory of Diophantine equations with a different flavor from the present course.

Now, we’ve discussed numbers having square-roots in some (or ) and not others. For example, I hope you can check that if and only if . But here is the quiz: Which rational numbers have square-roots in and *all* ?