A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers

Abstract: The Grunwald-Wang theorem for $n^{th}$ powers states that a rational number $a$ is an $n^{th}$ power in $\mathbb{Q}_{p}$ for almost every prime $p$ if and only if either $a$ is a perfect $n^{th}$ power in rationals or $8 \mid n$ and $a = 2^{\frac{n}{2}} \cdot b^{n}$ for some rational $b$. We will discuss an appropriate generalization of this theorem from a single rational number to a subset $A$ of rational numbers. Let $n$ be an odd number and $q$ be the smallest prime dividing $n$. A finite subset $A$ of rationals with cardinality $\leq q$ contains a $n^{th}$ power in $\mathbb{Q}_{p}$ for almost every prime $p$ if and only if $A$ contains a perfect $n^{th}$ power. For even $n$, the result is analogous - up to a short list of exceptions, as evident in the Grunwald-Wang theorem.

If time permits, we will also show that this generalization is optimal, i.e., for every $n \geq 2$, there are infinitely many subsets $A$ of rationals of cardinality $q+1$ that contain a $n^{th}$ power in $\mathbb{Q}_{p}$ for almost every prime $p$ but neither contain a perfect $n^{th}$ power nor fall under the finite list of exceptions.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar