Integers modulo n

In group theory

• Euler totient function: If $\phi :\mathbb{N}\to \mathbb{N}$ is Euler’s totient function, where $\phi (n)$ counts the number of $m<n$ which are coprime to $n$, then the group of units in $\mathbb{Z}/n\mathbb{Z}$ has cardinality $|U(n)|=\phi (n)$.

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In ring theory

• Algebraic rings: For all positive integers $n\ge 1$, the set $\mathbb{Z}/n\mathbb{Z}$ is a finite commutative ring. The group of units is the multiplicative group denoted $(\mathbb{Z}/n\mathbb{Z}{)}^{\ast }$, which is not a subring of $\mathbb{Z}/n\mathbb{Z}$.
• Algebraic fields: The ring $\mathbb{Z}/n\mathbb{Z}$ is a field iff $n$ is a prime number. This is often denoted ${\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$. In particular, this is one of the first examples of finite fields.
• Characteristics of rings: The characteristic of ${\mathbb{F}}_p$ is $p$, hence the characteristic of the polynomial ring ${\mathbb{F}}_p[x]$ is also $p$ since it contains ${\mathbb{F}}_p$ as a subring. Thus, ${\mathbb{F}}_p[x]$ is an example of an infinite integral domain with nonzero characteristic $p\ne 0$, and is not itself a field.
• Field of quotients of an integral domain: The Field of rational functions ${\mathbb{F}}_p$ (that is, the set of functions of the form $f/g$ where $f,g$ are polynomials with coefficients in ${\mathbb{F}}_p$) is an example of an infinite field with nonzero characteristic $p>0$. (This is because it contains a subring isomorphic to the polynomial ring ${\mathbb{F}}_p[x]$, which is infinite by the previous point).
• Prime (sub)fields: Every field of characteristic $p$ prime has a unique smallest subfield isomorphic to ${\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$, as a consequence of the induced homomorphism on its field of quotients. ${\mathbb{F}}_p$ itself is called a prime field since it contains no proper subfield.
• Ring homomorphisms and isomorphisms: The homomorphism $\phi :\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ is called reduction mod $n$.
• Prime and maximal ideals: Analogy between the quotient ring $F[x]/(f)$ and $\mathbb{Z}/n\mathbb{Z}$.

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Code snippets

\mathbb Z / n \mathbb Z

\mathbb F_p