Integers do not form Field

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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).

$\blacksquare$


Example

Take $2$, for example.

In the field of rational numbers, we have that $2 \times \dfrac 1 2 = 1$ and so $2$ has an inverse in $\Q$.

But that inverse is not in $\Z$.


Sources