Zorn’s Lemma

Adrien Douady () and Régine Douady
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Adrien Douady: Université Paris-Sud Orsay
Régine Douady: Université Paris Denis-Diderot

Chapter 1 in Algebra and Galois Theories, 2020, pp 1-19 from Springer

Abstract: Abstract The purpose we have in mind is infinite Galois theory. The notion of inverse limits of finite groups will be needed for this. By Tychonoff’s theorem, these are compact groups. The example in 6.4.6 shows that countable products are not sufficient. Tychonoff’s theorem is required in full, and hence so is Zorn’s lemma, and thus the axiom of choice. This chapter introduces the axiom of choice as well as the two results enabling its application: Zorn’s Lemma and Zermelo’s theorem. Both play more or less the same role, and either can usually be chosen. In our opinion, Zorn’s lemma is more powerful, while Zermelo’s theorem sometimes casts a more significant light.

Date: 2020
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DOI: 10.1007/978-3-030-32796-5_1

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