# Definition:Field of Integers Modulo Prime

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## 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

## Sources

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