Part III · Finite fields · Lesson 5

Which clocks are fields

The field axioms, and why ℤ/nℤ is a field exactly when n is prime.

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

You’ll learn

• The field axioms in plain terms: add, subtract, multiply, and divide by anything nonzero.
• In $\mathbb{Z}/6\mathbb{Z}$, $2 \cdot 3 = 0$ and $2$ has no inverse.
• For a prime $p$, every nonzero row of the multiplication table is a permutation, and the extended Euclidean algorithm finds each inverse. This field is $\mathrm{GF}(p)$.