DescriptionThis project will explore results that build on but go beyond what is covered in Elementary Number Theory II. Several results can be better understood by using a bit of algebra (groups and rings). We start with congruences modulo a power of a prime (Chapter 4 in Jones-Jones) and how this leads to \(p\)-adic numbers. We then study primitive roots from the point of view of generators of groups of the form \((\mathbb{Z}/n)^{\times}\). In ENT II we only have time to prove the existence of primitive roots modulo a prime, but here we will establish precisely for which natural numbers \(n\) a primitive root exists modulo \(n\) (Chapter 6 in Jones-Jones; Chapter 4 in Ireland-Rosen). Similarly, we then study quadratic residues modulo not only a prime, but a power of a prime and, ultimately, and arbitrary integer (Chapter 7 in Jones-Jones).Further topics for individual study depend on interest, but may include: Further theory of \(p\)-adic numbers, different proofs of quadratic reciprocity (e.g., using group theory), Quadratic Gauss Sums (Chapter 6 in Ireland-Rosen), Equations over Finite Fields (Chapter 10 in Ireland-Rosen). PrerequisitesElementary Number Theory II and Algebra II.Resources
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email: Alexander Stasinski