  We present a new structure theorem for finite fields of odd order that
relates multiplicative and additive structure in an interesting way. This
theorem has several applications, including an improved understanding of
Dickson and Chebyshev polynomials and some formulas with a number-theoretic
flavor. This paper is an abridged version of two math arXiv articles by the
author [arXiv:1707.06870, arXiv:1707.06877].
