Free groups and automorphism groups of infinite fields

2012/11/29 by Philipp Lücke, Saharon Shelah, Lücke, Philipp +1
Mathematics · #03E75 #12L99 (Secondary) #20E05 #20F29 (Primary) 03E35 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03E35 #msc:03E75 #msc:12L99 #msc:20E05 #msc:20F29

paper · pdf · doi:10.48550/arxiv.1211.6891

published as Forum Math. Sigma 2 (2014) e8, 18 · 13 pages

arxiv created 2013/01/18 · arxiv updated 2013/01/21

Abstract

Let λbe a cardinal with λ=λ^ℵ₀ and p be either 0 or a prime number. We show that there are fields K₀ and K₁ of cardinality λand characteristic p such that the automorphism group of K₀ is a free group of cardinality 2^λand the automorphism group of K₁ is a free abelian group of cardinality 2^λ. This partially answers a question from [8] and complements results from [15], [16] and [17]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that the existence of a cardinal λof uncountable cofinality with the property that there is no field of cardinality λwhose automorphism group is a free group of cardinality greater than λimplies the existence of large cardinals in certain inner models of set theory.

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