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Fields and Galois Theory

J.S. Milne

ℚ⁢[ζ]{\mathbb{Q}[\zeta]}ℚ⁢[ζ+ζ¯]{\mathbb{Q}[\zeta+\bar{\zeta}]}ℚ⁢[−7]{\mathbb{Q}[\sqrt{-7}]}ℚ{\mathbb{Q}}⟨σ3⟩\langle\sigma^{3}\rangle⟨σ2⟩\langle\sigma^{2}\rangle⟨σ⟩/⟨σ3⟩\langle\sigma\rangle/\langle\sigma^{3}\rangle⟨σ⟩/⟨σ2⟩\langle\sigma\rangle/\langle\sigma^{2}\rangle

Splitting field of X7−1X^{7}-1 over ℚ\mathbb{Q}.

ℚ⁢[ζ,α]{\mathbb{Q}[\zeta,\alpha]}ℚ⁢[ζ]{\mathbb{Q}[\zeta]}ℚ⁢[α]{\mathbb{Q}[\alpha]}ℚ{\mathbb{Q}}NNHHG/NG/N

Splitting field of X5−2X^{5}-2 over ℚ\mathbb{Q}.

Version 5.00
June 2021

Source: https://www.jmilne.org/math/CourseNotes/ft.html. Changes:

  • ⋄\diamond  
    ​

    Replace memoir with book class (in only).

  • ⋄\diamond  
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    Remove \documentclass from included file.

These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental extensions. The first six chapters form a standard course, and the final three chapters are more advanced.

BibTeX information
@misc{milneFT,
  author={Milne, James S.},
  title={Fields and Galois Theory (v5.00)},
  year={2021},
  note={Available at www.jmilne.org/math/},
  pages={142}
}

Please send comments and corrections to me at jmilne at umich.edu.

v2.01

(August 21, 1996). First version on the web.

v2.02

(May 27, 1998). Fixed many minor errors; 57 pages.

v3.00

(April 3, 2002). Revised notes; minor additions to text; added 82 exercises with solutions, an examination, and an index; 100 pages.

v4.00

(February 19, 2005). Revised notes; added proofs for Infinite Galois Extensions; expanded Transcendental Extensions; 107 pages.

v4.10

(January 22, 2008). Minor corrections and improvements; added proofs for Kummer theory; 111 pages.

v4.20

(February 11, 2008). Replaced Maple with PARI; 111 pages.

v4.30

(April 15, 2012). Minor fixes; added sections on étale algebras; 124 pages.

v4.50

(March 18, 2014). Added chapter on the Galois theory of étale algebras (Chapter 8); other improvements; numbering has changed; 138 pages.

v4.61

(April 2020). Minor fixes and additions; numbering little changed; 138 pages.

v5.00

(June 2021). First version available (with its source code) under a Creative Commons licence.

Version 5.0 is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0).



Copyright ©1996–2021 J.S. Milne.