College of Arts and Sciences
p-adic numbers, local-global principle, Hasse-Minkowski, Selmer
Barham, Benjamin, "Local-Global Principles for Diophantine Equations" (2020). Senior Honors Theses. 982.
The real number field, denoted ℝ, is the most well-known extension field of ℚ, the field of rational numbers, but it is not the only one. For each prime p, there exists an extension field ℚp of ℚ, and these fields, known as the p-adic fields, have some properties substantially different from ℝ. In this paper, we construct the p-adic numbers from the ground up and discuss the local-global principle, which concerns connections between solutions of equations found in ℚ and in ℚp. We state the Hasse-Minkowski theorem, which addresses a type of Diophantine equation to which the local-global principle applies, and conclude with a computation in which we apply the theorem followed by Selmer’s famous counterexample for cubic curves.