By symmetric function theorem in the title, the fundamental theorem of symmetric polynomials is meant: Any symmetric polynomial has a unique representation as a polynomial in the elementary symmetric polynomials.
Certainly, symmetric function theorem can be proved using Galois theory. But one is led to suspect that the connection is a little deeper than this simple one-way implication.
The line of thought arose after perusing Hecke's lectures in the theory of algebraic numbers. Usually early books on algebraic numbers look at all embeddings in the complex numbers to avoid mentioning the word ``automorphisms of fields'', in addition to the advantage of it being useful for proving various finiteness theorems. But this particular book bases the whole constructions on the symmetric function theorem and this baffles me to no end. As the simplest example, the proof that algebraic integers form a ring is done in this way. Certainly, I could grasp the proofs line-by-line; but the grasp of the big picture is still not there. Could somebody help? There is the unshakeable feeling that there is some deeper connections between the symmetric polynomial theorem and Galois theory or other studies of automorphims involving polynomials.