I've always been thrilled by the fact that the coefficients of a (monic) polynomial are obtained by taking the elementary symmetric functions in (minus) the roots of that polynomial:
$$\prod_\{i=1}^n (X+\alpha_i) = \sum_{k=0}^n (\sum_{i_1 < \cdots < i_k} \alpha_{i_1}\cdots \alpha_{i_k})X^{n-k}$$ A lot is built on this, I think. I'd like to explain the connection to automorphisms and fixed fields and how the roots of a polynomial are permuted by an automorphism that fixes the coefficient field of that polynomial. Then maybe mention the beginnings of Galois theory.