Introduction to number theory lecture 36 Kronecker symbol

Item: Introduction to number theory lecture 36 Kronecker symbol
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBHNumber theory

🎙 Richard E Borcherds 👥 82K 📅 March 21, 2022 ⏱ 21 min 👁 4K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

Kronecker symbolJacobi symbolLegendre symbolquadratic reciprocityDirichlet L-series

Summary

This lecture is part of a Berkeley undergraduate number theory course. It introduces the Kronecker symbol, a generalization of the Legendre and Jacobi symbols to all integer denominators. The speaker defines the symbol for various cases, including the special values for 2 and -1, and discusses its properties such as multiplicativity and periodicity. He highlights complications that arise when the denominator is zero or negative, and notes that the classical Kronecker symbol is often restricted to numerators congruent to 0 or 1 mod 4 and positive denominators. The lecture covers quadratic reciprocity for the Kronecker symbol, which involves a more complicated sign formula, and explains how to evaluate it by reducing to the Jacobi symbol. A table of values illustrates the periodicities and the differences between the three symbols. The speaker then applies the Kronecker symbol to imaginary quadratic fields, showing how it determines the splitting of primes and defines the Dirichlet L-series. As an example, he considers the Gaussian integers and derives the L-series for d = -4, which factorizes into a product over primes. Multiplying this L-series with the Riemann zeta function yields the zeta function of the field, and the speaker uses this to show that there are infinitely many primes congruent to 1 mod 4, a special case of Dirichlet’s theorem on primes in arithmetic progressions.

220 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a thorough and rigorous introduction to the Kronecker symbol, building on previously introduced concepts. The speaker carefully defines the symbol, discusses its properties, and highlights potential pitfalls. The argumentation is solid, with clear explanations and examples. The application to imaginary quadratic fields and Dirichlet L-series demonstrates the importance of the symbol and motivates its study. The presentation is well-paced and suitable for an advanced undergraduate audience.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on the textbook ‘An Introduction to the Theory of Numbers’ by Niven, Zuckerman, and Montgomery, a standard reference in number theory. The speaker is a well-known mathematician, and the content is mathematically accurate. The title accurately reflects the content. The lecture is part of a structured course, and the playlist link is provided for further context.

144 words

Title / Content Match

The title accurately describes the content: a lecture on the Kronecker symbol within a number theory course.

Quality & Reliability

9/10

Lecture by a renowned mathematician, part of a university course, based on a standard textbook. The content is rigorous and mathematically precise, with clear definitions and proofs. The presentation is well-structured and the speaker demonstrates deep expertise.

Key Moments

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

Cited Sources

Course playlist: Introduction to number theory — The lecture is part of this playlist, providing context and other lectures.

Concurring Sources

An Introduction to the Theory of Numbers — The textbook on which the lecture is based, providing a standard reference for the topic.

Contribution & Novelties

This lecture provides a clear and detailed exposition of the Kronecker symbol, a topic often treated briefly. It highlights the subtleties and restrictions needed to avoid inconsistencies, and demonstrates its utility in algebraic number theory and analytic number theory, particularly in the context of Dirichlet L-series and the proof of Dirichlet’s theorem. The presentation is accessible yet rigorous, making it a valuable resource for students.

Pour aller plus loin :

Kronecker symbol - Wikipedia — Overview and properties.
Dirichlet L-series - Wikipedia — Background on L-series and their role in number theory.
Dirichlet’s theorem on arithmetic progressions - Wikipedia — The theorem mentioned in the lecture.

106 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is both informative and reliable. The high technical level and quality of information are balanced by a moderate quantity of information, reflecting the focused scope of the lecture.

Reliability 9/10