Local solutions over $\mathbb{Q}_p$ but no solutions over $\mathbb{Q}$

Question

I was looking at a set of notes that states the equation $x^4-17=2y^2$ is solvable locally over $\mathbb{Q_p}$ for every $p$ , but is not solvable over $\mathbb{Q}$. Now, this is not a homework problem and is just for my own reading. Is there a simple way to see why this is the case? Is either direction "trivial"?

Answer 1

This one is hard. There is a complete discussion in Local Fields by Cassels. The following, Exercise 121 in Gouvea (yours is exercise 122) is easier: $$ (x^2-2)(x^2 - 17)(x^2-34)=0. $$ For one thing, there is just one variable.

Let's see, the Hasse-Minkowski principle says that this kind of example cannot happen for a quadratic form (form means homogeneous). But the same stuff does happen for Selmer's example: $$ 3 x^3 + 4 y^3 + 5 z^3 = 0, $$ where we require that $x,y,z$ not all be $0.$

I had been convinced for a long time that Fernando Q. Gouvea and Mariano Suárez-Alvarez were the same person. I met Gouvea in January. I've never met Mariano, but I now believe they are separate persons.