Reverse Mathematics and Algebraic Field Extensions

Abstract

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section §2, we show that WKL0 is equivalent to the ability to extend F-automorphisms of field extensions to automorphisms of 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. Reverse mathematics is a foundational program in which mathematical theorems are analyzed using a hierarchy of subsystems of second order arithmetic. This paper uses three such subsystems. The base system RCA0 includes 0 -IND (induction for 0 formulas) and set comprehension for 0 definable subsets of N. The stronger system WKL0 appends K¨ onig's theorem restricted to binary trees (subtrees of 2 <N ). The even stronger system ACA0 adds comprehension for arithmetically definable subsets of N. For a detailed formulation of these subsystems and related analysis of many mathematical theorems, see Simpson's book (15). Reverse mathematics of countable algebra, including topics from group theory, ring theory, and field theory, can be found in the paper of Friedman, Simpson, and Smith (4, 5). Further discussion appears throughout Simpson's book (15). A field is a set of natural numbers with operations and constants satisfying the field axioms. Field embeddings and isomorphisms can be defined as sets of (codes for) ordered pairs of field elements. Polynomials can be encoded by finite strings of coefficients, so polynomial rings are sets of (codes for) finite strings, with related ring operations. For details pertaining to any of these definitions, see either of the references above. Our study of fields begins in the next section with the definition of an algebraic field extension. To simplify the exposition in sections §1 through §3, we restrict our discussion to characteristic 0 fields. Consequently, in these sections all irreducible polynomials are separable. We indicate how to extend results of earlier sections to fields of other characteristics in section §6.