Reverse Mathematics and Algebraic Field Extensions

TL;DR

This paper applies reverse mathematics to analyze theorems about algebraic field extensions, establishing equivalences and exploring properties like automorphisms, embeddability, and Galois correspondence in the context of logical systems.

Contribution

It demonstrates the equivalence of certain algebraic extension theorems with weak König's lemma within reverse mathematics, providing a logical foundation for classical field theory results.

Findings

WKL_0 is equivalent to automorphism extension ability

Finitary conditions for embeddability are characterized

Galois correspondence theorems are extended to infinite extensions

Abstract

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.