Lift roots of a polynomial from $\mathbb{F}_p$ to $\mathbb{Q}_p$ I was trying to prove a counterexample to Hasse's principle. To do this, I was trying to prove that the equation $x^4-17=2y^2$ has no solutions on $\mathbb{Q}$ but has solutions in $\mathbb{Q}_p$ for every prime $p$.
I was able to prove that the equation has no solutions on $\mathbb{Q}$, but I was not able to do the second part. What I've done was to prove (using Riemman-Hurwitz) that  $x^4-17=2y^2$ defines an elliptic curve and thus Hasse's theorem gives us that the curve is non empty over $\mathbb{F}_p$ (i.e. the equation has at least one solution over $\mathbb{F}_p$). The cases $p=2,17$ are special, but these are easily checked.
From here I know that I have to use Hensel's lemma to lift the solution to $\mathbb{Q}_p$. This is the Hensel's lemma I was trying to use (maybe it is not the correct form of Hensel's lemma):
Let $(K,|\cdot|)$ be a complete valuation field, $f\in R[X]$ and $a_0\in R$ such that $|f(a_0)|<|f'(a_0)|^2$, where $R=\lbrace x\in K\mid|x|\leq 1\rbrace$. Then, there exists a unique $a\in R$ such that $f(a)=0$ and $a-a_0\in\mathcal{M}$, where $\mathcal{M}=\lbrace x\in K\mid|x|< 1\rbrace$.
I'm not sure how to use this, since the theorem involves a polynomial in $x$, but I'm working with a polynomial in $x$ and $y$. Also, I'm not sure how to relate roots in $\mathbb{F}_p$ and roots in $\mathbb{Q}_p$. I know that the residual field of $\mathbb{Q}_p$ is $\mathbb{F}_p$ (i.e. $R/\mathcal{M}\cong\mathbb{F}_p$) but I'm not sure how this relates with Hensel's lemma.
Any hints or help will be thanked.
 A: Let me just fill in the details of KCd's comment for completeness. First assume $p\neq 2,17$, and let $(a+p\mathbb{Z},b+p\mathbb{Z})$ be a root of our polynomial in $\mathbb{F}_p$. Then at least one of $a$ and $b$ is non-zero mod $p$; otherwise $17$ would be zero mod $p$, a contradiction. So we have two cases.
First suppose $a\notin p\mathbb{Z}$. Consider the polynomial $g(x)=x^4-(17+2b^2)$. Then $g(a)\equiv 0$ mod $p$ and $g'(a)=4a\not\equiv 0$ mod $p$, so by Hensel's lemma there is a root $a'$ of $g$ in $\mathbb{Q}_p$. Now $(a',b)$ is a $\mathbb{Q}_p$-rational point of our curve.
Now suppose $b\notin p\mathbb{Z}$. Then consider the polynomial $h(y)=2y^2+(17-a^4)$. Then $h(b)\equiv 0$ mod $p$ and $h'(b)=4b\not\equiv 0$ mod $p$, so by Hensel's lemma there is a root $b'$ of $g$ in $\mathbb{Q}_p$. Now $(a,b')$ is a $\mathbb{Q}_p$-rational point of our curve. Either way, we are done.

For the case $p=17$, you can use almost an identical argument as above; we just need to find an non-zero $\mathbb{F}_{17}$-rational point of the polynomial to work. (Why does this suffice?) For example, take $(1+17\mathbb{Z},3+17\mathbb{Z})$. Finally, for the case $p=2$, it would suffice to show that $17$ is a fourth power in $\mathbb{Q}_2$, and by Hensel's lemma it suffices for this to show that $17$ is a fourth power mod $2^5$. (Why?) But $3^4=81$ is congruent to $17$ mod $2^5$, as needed.
