Characteristic of an Integral Domain

Hi Testere, let's look at this abstract algebra proof together. We're going to prove that the characteristic of an integral domain D must be either zero or a prime number.

First, let's recall what the characteristic and an integral domain are.

Suppose the characteristic of D is not zero. We want to show it must be a prime number. To do this, we'll use a proof by contradiction.

Let's assume the opposite: that $n$ is a composite number.

This means we can write n as the product of two smaller integers, a and b, where both are greater than one and less than n.

Since the characteristic of the domain is n, we know that n times the multiplicative identity, one, equals zero.

Now, we can substitute our product a times b in place of n.

By the properties of ring multiplication and the identity element, we can rewrite this product. Note that one squared is just one.

Let's look closely at this equation. We have a product of two elements from our domain equal to zero.