Integers do not form Field

Corollary of Invertible Integers under Multiplication
The integers $\struct {\Z, +, \times}$ do not form a field.

Proof
For $\struct {\Z, +, \times}$ to be a field, it would require that all elements of $\Z$ have an inverse.

However, from Invertible Integers under Multiplication, only $1$ and $-1$ have inverses (each other).