Reverse Mathematics and Algebraic Field Extensions

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section 2, we show that $\mathsf{WKL}_0$ is equivalent to the ability to extend $F$-automorphisms of field extensions to automorphisms of $\bar{F}$, the algebraic closure of $F$. Section 3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section 4, and the Galois correspondence theorems for infinite field extensions are treated in section 5.Comment: 25 page

The reverse mathematics of the Tietze extension theorem

We prove that several versions of the Tietze extension theorem for functions with moduli of uniform continuity are equivalent to WKL0 over RCA0. This con rms a conjecture of Giusto and Simpson [3] that was also phrased as a question in Montalb an's Open questions in reverse mathematics [6]

Local-global principles for Galois cohomology

This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, Z/mZ(n-1))$, for all $n>1$. This is motivated by work of Kato and others, where such principles were shown in related cases for $n=3$. Using our results in combination with cohomological invariants, we obtain local-global principles for torsors and related algebraic structures over $F$. Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.Comment: 32 pages. Some changes of notation. Statement of Lemma 2.4.4 corrected. Lemma 3.3.2 strengthened and made a proposition. Some proofs modified to fix or clarify specific points or to streamline the presentatio

Representations and the foundations of mathematics

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.Comment: 21 pages, one figure. Uses the same 'preliminaries' section as arXiv:1910.02489, arXiv:1908.05677, arXiv:1908.05676, arXiv:1905.0405

A definable henselian valuation with high quantifier complexity

We give an example of a parameter-free definable henselian valuation ring which is neither definable by a parameter-free $\forall\exists$-formula nor by a parameter-free $\exists\forall$-formula in the language of rings. This answers a question of Prestel.Comment: 6 page