Field of Integers Modulo Prime is Prime Field

Theorem

Let $p$ be a prime number.

Let $\left({\Z_p, +, \times}\right)$‎ be the field of integers modulo $p$.

Then $\left({\Z_p, +, \times}\right)$‎ is a prime field.

Proof

If $\left({F, +, \times}\right)$ is a subfield of $\left({\Z_p, +, \times}\right)$, then $\left({F, +}\right)$ is a subgroup of $\left({\Z_p, +}\right)$.

But from Prime Group has No Proper Subgroups, $\left({\Z_p, +}\right)$ has no proper subgroup except the Trivial Group.

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

$\blacksquare$

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.2$: Example $2$