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The key result is the existence of primitive roots modulo a prime. This theorem was used by mathematicians before Gauss but he was the first to give a proof.The existence of primitive roots is equivalent to the fact that $U(\mathbb{Z} / p \mathbb{Z})$ is a cyclic group when $p$ is a prime. Using this fact we shall find an explicit description of the group $U(\mathbb{Z} / n \mathbb{Z})$ for arbitrary $n$.
高斯是通过一个中间量，也就是所谓的order來证明这个结构的，他发现可以这个群里面阶大于等于特定值的元素个数是可以估计上界的，这由唯一分解保证（因为唯一分解导致代数基本定理）。另一方面作为整体是有上界的，这两件事情夹逼就可以得到想要的结论，多项式在这中间起到过渡作用成为链接物理空间和谐波空间的桥梁。
The polynomial method is a relatively new algebraic tool (and philosophy) that has over the past ten years enabled researchers to settle several long-standing open problems arising from diverse areas such as Combinatorial and Finite Geometry, Additive Combinatorics, Number theory, and so on. I shall in this note, provide an introduction to what this method is all about, and as an attempt to expound on this new technique, go over four problems in which substantial progress happened from a prior state of virtual hopelessness. In particular, we shall consider the following:
- Dvir’s solution of the Finite Kakeya Conjecture,
- Guth-Katz’ solution of the Joints’ Problem,
- The Cap-set problem and the work of Ellenberg-Gisjwijt, and
- A function field analogue of Sárközy’s theorem, due to Green.
下面是一些经典的The Structure of $U(\mathbb{Z} / n \mathbb{Z})$题目
Key techniques or knowledge
understanding of Möbius function.
Key techniques or knowledge
Guass’s lemma, Fermat’s little theorem.
Key techniques or knowledge
Wilson’s theorem. This involves Euler’s theorem
Key techniques or knowledge
applying Euclid’s algorithm
Key techniques or knowledge
the property of order of an element
Key techniques or knowledge
the structure theorem for finitely generated abelian groups.
Key techniques or knowledge
the Chinese remainder theorem.
