Part III · Finite fields · Lesson 6

Every field with p elements is ℤ/pℤ

Characteristic and the prime subfield: there is only one field of each prime order.

This lesson is planned. Here is what it will cover.

You’ll learn

• Adding $1$ to itself, $1, 1 + 1, 1 + 1 + 1, \dots$, first returns to $0$ after a prime number $p$ of steps: the characteristic.
• The multiples of $1$ form a copy of $\mathbb{Z}/p\mathbb{Z}$, the prime subfield; when the field has $p$ elements, $k \mapsto k \cdot 1$ is an isomorphism.
• Every finite field is a vector space over its prime subfield, so its size is $p^k$.