A quiz about p-adic numbers

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 {\bf Q}_p (or {\bf R}) and not others. For example, I hope you can check that \sqrt{-1}\in {\bf Q}_p if and only if p \equiv 1 \mod 4. But here is the quiz: Which rational numbers have square-roots in {\bf R} and *all* {\bf Q}_p?


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