The product of two nonzero field elements is nonzero ↩

Metadata

shorthands: {}
aliases: [No divisors of zero]
created: 2021-11-05 23:19:53
modified: 2022-01-10 04:13:04

Statement:
Let

be a field. If

and

, then

.
Equivalently:

Proof

Let's prove the logically equivalent contrapositive, i.e. .

Because , then it has an inverse and .

With this: . QED

This theorem describes why zero does not have a divisor. We cannot have , , but . So a field has no divisors of zero.