Galois theory: Norm and trace

Item: Galois theory: Norm and trace
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBFAlgebra

🎙 Richard E Borcherds 👥 82K 📅 January 15, 2021 ⏱ 24 min 👁 7K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

normtracefield extensionalgebraic integersGalois theory

Summary

This lecture from a graduate course on Galois theory introduces the concepts of norm and trace for finite field extensions. The speaker defines the trace and norm of an element as the trace and determinant of the linear transformation of multiplication by that element. He derives formulas for the trace and norm in terms of the minimal polynomial, noting the dependence on the field extension. He discusses the absolute trace and norm, and their limitations. Examples are given for extensions like R over C and Q over Q(i), illustrating the quotient of multiplicative groups. The norm map for finite fields is shown to be surjective. The lecture then applies norm and trace to find rings of algebraic integers in quadratic fields, deriving conditions for when elements are algebraic integers. The lecture concludes by hinting at the use of trace to define a bilinear form in the next lecture.

148 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous introduction to norm and trace, with well-chosen examples that illustrate the concepts and their applications. The argumentation is solid, building from definitions to formulas and then to applications. The speaker explains the reasoning behind each step, making the material accessible to graduate students. The examples, such as the norm map for Q(i) and the surjectivity for finite fields, are particularly illuminating. The application to finding algebraic integers in quadratic fields demonstrates the practical utility of the concepts.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is mathematically rigorous, with careful definitions and proofs. The speaker does not cite external sources, but the content is standard and well-established. The title accurately reflects the content. The lecture is part of a structured course, and the speaker’s expertise is evident. No comments were provided, so no analysis of public reception is possible.

155 words

Title / Content Match

The title accurately reflects the content, which focuses on the norm and trace in field extensions.

Quality & Reliability

9/10

Lecture by a renowned mathematician, rigorous definitions and proofs, examples and applications, no obvious errors.

Key Moments

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

Contribution & Novelties

The lecture provides a clear and rigorous exposition of norm and trace, with insightful examples and applications. It bridges abstract algebra and number theory, showing how these concepts are used in practice. The discussion of the quotient k*/N(m*) and the surjectivity for finite fields are particularly valuable. The application to algebraic integers in quadratic fields is a concrete demonstration of the utility of norm and trace.

Pour aller plus loin :

Algebraic number theory — Provides background on algebraic integers and number fields.
Field extension — Basic definitions and properties of field extensions.
Norm (field theory) — Wikipedia article on the norm in field theory.
Trace (field theory) — Wikipedia article on the trace in field theory.

117 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is both informative and reliable, with a strong technical level. The balance between quantity and quality of information is excellent, making it a valuable resource for graduate students.

Reliability 9/10