Solubility of a resultant equation and applications
Abstract
The large sieve is used to estimate the density of integral quadratic polynomials Q, such that there exists an odd degree integral polynomial which has resultant 1 with Q. Given a monic integral polynomial R of odd degree, this is used to show that for almost all integral quadratic polynomials Q, there exists a prime p such that Q and R share a common root in the algebraic closure of the finite field with p elements. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.