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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"?

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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.

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  • $\begingroup$ Just to add more detail (paraphrasing Gouvea): this is on page 57 of Local Fields. A proof using algebraic number theory is also outlined by Cassels in Lectures on Elliptic Curves page 88. $\endgroup$
    – Deven Ware
    Nov 27, 2013 at 5:58
  • $\begingroup$ OK thank you. I will take a look at these. I wanted to make sure I was not seeing something trivial. $\endgroup$
    – LASV
    Nov 27, 2013 at 5:59

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