transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
For $n \in \mathbb{N}$, the group of units of the ring of integers modulo n
is typically called the multiplicative group of integers modulo $n$.
This consists of all those elements $k \in \mathbb{Z}/n$ which are represented by coprime integers to $n$. It is typically denoted $\mathbb{Z}/n ^\ast$.
The cardinality ${|(\mathbb{Z}/n)^\times|}$ is often denoted $\phi(n)$ (after Leonhard Euler), and $\phi$ is known as the Euler totient function.
The function $\phi$ is multiplicative in the standard number theory sense: $\phi(m n) = \phi(m)\phi(n)$ is $m$ and $n$ are coprime. This is a corollary of the Chinese remainder theorem?, which asserts that the canonical ring map
is an isomorphism if $m, n$ are coprime. Thus, if $n = p_1^{r_1} \ldots p_k^{r_k}$ is the prime factorization of $n$, we have
Furthermore, as $\mathbb{Z}/p^r$ is a local ring with maximal ideal $p(\mathbb{Z}/p^r)$, the cardinality of the group of units is
Wikipedia, Multiplicative group of integers modulo n
Wikipedia, Euler’s totient function
Last revised on December 28, 2022 at 12:18:42. See the history of this page for a list of all contributions to it.