# Example to show that a factor ring of an integral domain may be a field.

I have seen this question asked elsewhere on here, but with a different direction than the one I want to go. The question is asking me to give an example to show that a factor ring of an integral domain may be a field. The example I was thinking of was $$\Bbb Z/(p\Bbb Z)$$, and I have convinced myself that this is a field but am not sure the exact math way that I would prove that this is a field using theorems. Could someone give me a pointer on this?

The theorems you're looking for are:

• if $$R$$ is a commutative ring with $$1$$ and $$I$$ is an ideal of $$R$$, then $$R/I$$ is a field if and only if $$I$$ is maximal.

• if $$R$$ is a commutative ring with $$1$$ and $$I$$ is an ideal of $$R$$, then $$R/I$$ is an integral domain if and only if $$I$$ is prime.

You can find proofs in every algebra book (e.g. Dummit & Foote, Lang, Hungerford, Fraleigh, etc) or even in questions in this website.

So, since $$\Bbb Z$$ is commutative with $$1$$ and $$p\Bbb Z$$ is a maximal ideal for $$p$$ prime, $$\Bbb Z/p\Bbb Z$$ is a field.

• Good and complete. Jun 2, 2019 at 23:39

For any prime

$$p \in \Bbb P \subset \Bbb Z, \tag 1$$

it is easy to see that

$$0 < k < p \Longrightarrow \gcd(k, p) = 1; \tag 2$$

indeed,

$$p \not \mid k, \; 0 < k < p, \tag 3$$

and since the only divisors of $$p$$ are $$1$$ and $$p$$ itself, the only common divisor of $$p$$ and $$k$$ is $$1$$; hence (2).

We next exploit the well-known Bezout's identity, that is

$$\exists a, b \in \Bbb Z, \; ap + bk = \gcd(p, k) = 1; \tag 4$$

when we reduce this equation modulo $$p$$, that is, when we form the cosets of $$(p) \subset \Bbb Z$$, we obtain

$$(b +(p))(k + (p)) = (a + (p))(p + (p)) + (b +(p))(k + (p))$$ $$= (ap + (p)) + (bk + (p)) = (ap + bk) + (p) = 1 + (p), \tag 5$$

since

$$(a + (p))(p + (p)) = (a + (p))(0 + (p)) \equiv 0 \mod p; \tag 6$$

(5) shows that every $$k + (p)$$ has a multiplicative inverse $$b + (p)$$ in the commutative ring $$\Bbb Z/(p)$$; hence $$\Bbb Z / (p)$$ is a field.