Is it true that if an equation has solutions in $\mathbb Q^2$, it has a solution in $\mathbb Q_p^2$ for all primes $p$? For example, if $f(a, b) = a^2 - 2b^2$, the only solution of $f(a,b) = 0$ in $\mathbb Q^2$ is $(0,0)$. For $p\neq 2$, this gives a solution in $\mathbb Q_p^2$ using Hensel's lemma. However, for $p = 2$, Hensel's lemma fails. Does this mean that there is no solution of $f(a,b) = 0$ in $\mathbb Q_2^2$? I'm confused about this because there exists an embedding $\mathbb Q\hookrightarrow\mathbb Q_p$. Does this embedding not preserve equations?
How can we show that some equation does not have any solutions in $\mathbb Q_p^2$?
2026-04-01 17:04:35.1775063075