Field of Integers Modulo Prime is Prime Field

Theorem

Let $p$ be a prime number.

Let $\struct {\Z_p, +, \times}$‎ be the field of integers modulo $p$.

Then $\struct {\Z_p, +, \times}$‎ is a prime field.

Proof

If $\struct {F, +, \times}$ is a subfield of $\struct {\Z_p, +, \times}$, then $\struct {F, +}$ is a subgroup of $\struct {\Z_p, +}$.

But from Prime Group has no Proper Subgroups, $\struct {\Z_p, +}$ has no proper subgroup except the trivial group.

Hence $F = \Z_p$ and so follows the result.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties: Example $2$