  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.
