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 $\bar F$, 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.