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Reverse mathematics and algebraic field extensions. (English) Zbl 1308.03038
This paper analyzes algebraic field extensions from the perspective of reverse mathematics. For this, fields are, as usual in reverse mathematics, coded on \(\mathbb{N}\) and thus assumed to be countable.
The following are the main results of the paper:
Let \(F\) be a field and \(K\), \(J\) be two isomorphic field extensions. Then, \(\mathsf{WKL}_0\) is equivalent to the statement that the algebraic closures \(\bar{K}\), \(\bar{J}\) are isomorphic.
Let \(F\) be a field and \(K\), \(J\) algebraic extensions of \(F\). Then, \(\mathsf{ACA}_0\) is equivalent to the statement that if for all \(k\in K\) the field \(F(k)\) embeds into \(J\) fixing \(F\), then \(K\) is embedded into \(J\) fixing \(F\).
Similar characterizations are given for Galois extensions.
03B30 Foundations of classical theories (including reverse mathematics)
12F05 Algebraic field extensions
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