Hasse Principle -- Maksym Fedorchuk
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Hasse principle posits that a homogeneous Diophantine equation has a
non-trivial solution if and only if it has a non-trivial real solution
and a non-trivial p-adic solution for all prime p.  We'll discuss
Hasse-Minkowski theorem, which establishes Hasse principle for
quadratic equations, and, if time permits, counterexamples to Hasse
principle in degree 3.