We present a new structure theorem for finite fields of odd order that relates multiplicative and additive properties 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 articles by the author.

中文翻译：

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有限域的结构定理

我们提出了一种奇数阶有限域的新结构定理，该定理以一种有趣的方式关联了乘法和加性。该定理具有多种应用，包括对Dickson和Chebyshev多项式的更好理解以及一些具有数论风格的公式。本文是作者撰写的两篇文章的节略版。