Introduction to number theory lecture 35 Jacobi symbol

Introduction to number theory lecture 35 Jacobi symbol

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Richard E Borcherds 👥 82K 📅 March 19, 2022 ⏱ 38 min 👁 6K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

Jacobi symbolLegendre symbolquadratic reciprocityZolotarevprimality test

Summary

This lecture from a Berkeley undergraduate number theory course introduces the Jacobi symbol, a generalization of the Legendre symbol to composite odd denominators. The instructor defines the Jacobi symbol as a product of Legendre symbols over prime factors and proves its basic properties, including multiplicativity in both arguments, formulas for (-1/b) and (2/b), periodicity, and the law of quadratic reciprocity. He emphasizes a key warning: a Jacobi symbol of +1 does not imply the numerator is a quadratic residue modulo the denominator, illustrating with 2/15 = +1. The lecture then demonstrates how the Jacobi symbol can be computed efficiently using an algorithm analogous to the Euclidean algorithm, avoiding factorization of the numerator. This efficiency is applied to a primality test based on Euler’s criterion, showing that 561 is not prime. Finally, the lecture presents Zolotarev’s definition of the Jacobi symbol as the sign of a permutation, proving its equivalence and discussing subtleties with the sign of product permutations.

158 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a thorough and rigorous introduction to the Jacobi symbol, building on the Legendre symbol. The proofs are clear and complete, with careful attention to edge cases. The computational algorithm is well-motivated and demonstrated with examples. The application to primality testing is relevant and illustrates the practical importance of the Jacobi symbol. The presentation of Zolotarev’s definition adds depth and connects to group theory.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is mathematically rigorous, with all claims proven or left as straightforward exercises. The instructor references the standard textbook by Niven, Zuckerman, and Montgomery. The title accurately reflects the content. No external sources are cited beyond the textbook and course playlist.

123 words

Title / Content Match

The title accurately reflects the content, which is a lecture on the Jacobi symbol.

Quality & Reliability

9/10

Lecture by a renowned mathematician, rigorous proofs, references to standard textbook, no apparent errors.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Concurring Sources

  • An Introduction to the Theory of Numbers — Textbook referenced in the lecture

Contribution & Novelties

The lecture provides a clear and rigorous exposition of the Jacobi symbol, including its efficient computation and Zolotarev’s definition. It highlights the key difference from the Legendre symbol and its application to primality testing.

Pour aller plus loin :

76 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a well-rounded and reliable lecture. The slightly lower technical level reflects the undergraduate audience, but the content is still rigorous.

Reliability 9/10