field3

Field3, also known as the field with three elements, is the finite field consisting of the three integers 0, 1, and 2 under ordinary addition and multiplication performed modulo 3. It is the smallest nontrivial finite field and a prime field, often denoted GF(3) or F3.

The elements are 0, 1, and 2. Under addition modulo 3, 0 acts as the additive identity, and every element has an additive inverse: 0 is its own inverse, 1’s inverse is 2, and 2’s inverse is 1. Under multiplication modulo 3, 1 acts as the multiplicative identity, and every nonzero element has a multiplicative inverse: 1 × 1 ≡ 1, 2 × 2 ≡ 1. The nonzero elements form a multiplicative group of order 2. The field has characteristic 3, meaning 1 + 1 + 1 ≡ 0.

Field3 is isomorphic to the quotient ring Z/3Z and is used as a simple example of a finite field in undergraduate algebra. It serves as a null example for studying field axioms, modular arithmetic, and polynomial arithmetic modulo 3. In higher mathematics, larger finite fields GF(p^n) are built from prime fields like Field3 as foundational components.

In practice, Field3 appears in teaching, in small-scale coding theory demonstrations, and as a stepping stone to understanding more complex finite fields. It is often denoted F3 or GF(3) in mathematical literature.