Galois theory: Examples of Galois extensions

Galois theory: Examples of Galois extensions

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematicsPBFAlgebra
🎙 Richard E Borcherds 👥 82K 📅 January 3, 2021 ⏱ 20 min 👁 14K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

Galois extensionGalois groupSubfieldSubgroupCyclotomic field

Summary

This lecture, part of an online graduate course on Galois theory, presents several examples of Galois extensions and explicitly works out the correspondence between subfields and subgroups. The instructor begins with a trivial example (R⊂C) to illustrate the basic principle that larger fields correspond to smaller subgroups. He then moves to a more substantial example: the splitting field of x^3-2 over Q, which has Galois group S3. He draws the lattice of subfields and subgroups, showing how they are dual. Next, he discusses finite fields, showing that the Galois group of F_{16} over F_2 is cyclic of order 4, generated by the Frobenius automorphism. He then considers the cyclotomic field Q(ζ_7), where ζ_7 is a primitive 7th root of unity. The Galois group is cyclic of order 6, and he explicitly identifies the subfields corresponding to each subgroup, including the real subfield Q(ζ_7+ζ_7^{-1}) and the quadratic subfield Q(√-7). The lecture concludes by foreshadowing the proof of the fundamental theorem of Galois theory in the next lecture.

166 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides high-value information by illustrating abstract Galois theory with concrete, well-chosen examples. The argumentation is solid: each example is carefully constructed, and the correspondence is explicitly computed. The instructor explains the reasoning behind each step, making the material accessible while maintaining rigor. The examples progress from simple to more complex, helping to build intuition. The use of lattices and explicit computations enhances understanding. The lecture is particularly valuable for its demonstration of how to find subfields corresponding to subgroups, such as the quadratic subfield in the cyclotomic example.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, with clear definitions and proofs. The instructor relies on standard results from field theory and group theory, and the examples are accurate. The title accurately reflects the content, which is focused on examples of Galois extensions. No external sources are cited, but the lecture is based on well-established mathematical knowledge. The presentation is clear and well-structured, with no apparent errors. The instructor’s expertise is evident, and the lecture is suitable for a graduate-level audience.

184 words

Title / Content Match

The title accurately reflects the content, which focuses on examples of Galois extensions and the Galois correspondence.

Quality & Reliability

9/10

Lecture by a renowned mathematician, rigorous and clear, with explicit examples and proofs. The content is standard and well-established, and the presentation is accurate.

Key Moments

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

Contribution & Novelties

This lecture provides a clear and detailed exposition of Galois correspondence through concrete examples. It is particularly valuable for its explicit computation of subfields corresponding to subgroups, such as the quadratic subfield in the cyclotomic example. The lecture also highlights the duality between subfields and subgroups, and the importance of normal subgroups corresponding to normal extensions.

Pour aller plus loin :

93 words

Radar Profile

The radar profile shows high scores in all dimensions, with particularly strong quality of information and reliability. The lecture is technically solid and provides substantial content, making it an excellent resource for learning Galois theory.

Reliability 9/10