Field with two elements (${\mathbb{F}}_2$)

The field with two elements is ${\mathbb{F}}_2=\mathbb{Z}/2\mathbb{Z}$, with elements $0,1$ and arithmetic modulo two. Its only nonzero element is its own multiplicative inverse; in particular, $1+1=0$ and $-1=1$. These rules satisfy the field axioms.

Interpreting $0,1$ as bits, addition is exclusive-or and multiplication is logical conjunction. For instance, $(1,0,1)+(1,1,0)=(0,1,1)$ in the vector space ${\mathbb{F}}_2^3$. This makes linear algebra over the field with two elements useful for binary linear codes and parity constraints.