Definition:Field of Integers Modulo Prime

Definition

Let $p \in \Bbb P$ be a prime number.

Let $\Z_p$ be the set of integers modulo $p$.

Let $+_p$ and $\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively.

The algebraic structure $\struct {\Z_p, +_p, \times_p}$ is the field of integers modulo $p$.

Also denoted as

When the operations are understood to be $+_p$ and $\times_p$, it is usual to use just $\Z_p$ to denote the field of integers modulo $p$.

The notation $\Z / p$ and $\Z / p \Z$ are also seen, deriving from Quotient Ring of Integers by Integer Multiples.

Some sources use the notation $\map {\mathrm {GF} } p$, in light of the fact that $\struct {\Z_p, +_p, \times_p}$ is a Galois field.

Also see

• Ring of Integers Modulo Prime is Field, where it is shown that:

$\struct {\Z_p, +_p, \times_p}$ is a field with unity $\eqclass 1 p$

the zero of $\struct {\Z_p, +_p, \times_p}$ is $\eqclass 0 p$.

• Definition:Galois Field, of which this is an example

• Results about the field of integers modulo prime can be found here.

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts