Reduction to Primitive Pythagorean Triples Diophantine Equations Convergence of $\exp$ and $\log$ Local Fields Every Dirichlet Character Is Induced by a Unique Primitive Character Analytic Number Theory Basic Algebraic Properties of the Hilbert Symbol Diophantine Equations Completion Formula for a Short Sum Modulo $q$ Analytic Number Theory Ring Isomorphism Form of CRT Elementary Number Theory Existence Theorem (Local) Local Fields Affine Point with Zero $y$-Coordinate Is Two-Torsion on a Short Weierstrass Elliptic Curve Number Theory Number Theory Area Analytic Number Theory Subarea