Document Type
Lecture
Publication Date
10-20-2023
Abstract
The Grunwald-Wang theorem for nth powers states that a rational number a is an nth power in ℚp for almost every prime p if and only if either a is a perfect nth power in rationals or 8 | n and a = 2^(n/2) b^n for some rational b. In this talk, we will present a generalization of a Grunwald-Wang theorem, from a single integer a to a subset A of rational numbers. More specifically, let q be the smallest prime dividing the natural number n ≥ 2. A finite subset A of rationals with cardinality ≥ q contains an nth power in ℚp for almost every prime p if and only if either A contains an nth power in rationals or n is even and A is a two-element subset of a certain form. If time permits, we will also show that our generalization is optimal, i.e., for every n ≥ 2, there are infinitely many subsets A of rationals of cardinality q + 1 that contain an nth power in ℚp for almost every prime p but neither contain a perfect nth power in rationals nor contain a two-element subset of the above kind when n is even.
Relational Format
presentation
Recommended Citation
Mishra, Bhawesh, "A Generalization of the Grunwald-Wang Theorem" (2023). Algebra/Number Theory Seminar. 1.
https://egrove.olemiss.edu/math_algebra_number_theory/1
