## Section 5.2 Integral Domains and Fields

Let us briefly recall some definitions. If \(R\) is a ring and \(r\) is a nonzero element in \(R\text{,}\) then \(r\) is said to be a zero divisor if there is some nonzero element \(s \in R\) such that \(rs = 0\text{.}\) A commutative ring with identity is said to be an integral domain if it has no zero divisors. If an element \(a\) in a ring \(R\) with identity has a multiplicative inverse, we say that \(a\) is a unit. If every nonzero element in a ring \(R\) is a unit, then \(R\) is called a division ring. A commutative division ring is called a field.

###### Example 5.12.

If \(i^2 = -1\text{,}\) then the set \({\mathbb Z}[ i ] = \{ m + ni : m, n \in {\mathbb Z} \}\) forms a ring known as the Gaussian integers. It is easily seen that the Gaussian integers are a subring of the complex numbers since they are closed under addition and multiplication. Let \(\alpha = a + bi\) be a unit in \({\mathbb Z}[ i ]\text{.}\) Then \(\overline{\alpha} = a - bi\) is also a unit since if \(\alpha \beta = 1\text{,}\) then \(\overline{\alpha} \overline{\beta} = 1\text{.}\) If \(\beta = c + di\text{,}\) then

Therefore, \(a^2 + b^2\) must either be \(1\) or \(-1\text{;}\) or, equivalently, \(a + bi = \pm 1\) or \(a + bi = \pm i\text{.}\) Therefore, units of this ring are \(\pm 1\) and \(\pm i\text{;}\) hence, the Gaussian integers are not a field. We will leave it as an exercise to prove that the Gaussian integers are an integral domain.

###### Example 5.13.

The set of matrices

with entries in \({\mathbb Z}_2\) forms a field.

###### Example 5.14.

The set \({\mathbb Q}( \sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}\) is a field. The inverse of an element \(a + b \sqrt{2}\) in \({\mathbb Q}( \sqrt{2}\, )\) is

We have the following alternative characterization of integral domains.

###### Proposition 5.15. Cancellation Law.

Let \(D\) be a commutative ring with identity. Then \(D\) is an integral domain if and only if for all nonzero elements \(a \in D\) with \(ab = ac\text{,}\) we have \(b=c\text{.}\)

###### Proof.

Let \(D\) be an integral domain. Then \(D\) has no zero divisors. Let \(ab = ac\) with \(a \neq 0\text{.}\) Then \(a(b - c) =0\text{.}\) Hence, \(b - c = 0\) and \(b = c\text{.}\)

Conversely, let us suppose that cancellation is possible in \(D\text{.}\) That is, suppose that \(ab = ac\) implies \(b=c\text{.}\) Let \(ab = 0\text{.}\) If \(a \neq 0\text{,}\) then \(ab = a 0\) or \(b=0\text{.}\) Therefore, \(a\) cannot be a zero divisor.

The following surprising theorem is due to Wedderburn.

###### Theorem 5.16.

Every finite integral domain is a field.

###### Proof.

Let \(D\) be a finite integral domain and \(D^\ast\) be the set of nonzero elements of \(D\text{.}\) We must show that every element in \(D^*\) has an inverse. For each \(a \in D^\ast\) we can define a map \(\lambda_a : D^\ast \rightarrow D^\ast\) by \(\lambda_a(d) = ad\text{.}\) This map makes sense, because if \(a \neq 0\) and \(d \neq 0\text{,}\) then \(ad \neq 0\text{.}\) The map \(\lambda_a\) is one-to-one, since for \(d_1, d_2 \in D^*\text{,}\)

implies \(d_1 = d_2\) by left cancellation. Since \(D^\ast\) is a finite set, the map \(\lambda_a\) must also be onto; hence, for some \(d \in D^\ast\text{,}\) \(\lambda_a(d) = ad = 1\text{.}\) Therefore, \(a\) has a left inverse. Since \(D\) is commutative, \(d\) must also be a right inverse for \(a\text{.}\) Consequently, \(D\) is a field.

For any nonnegative integer \(n\) and any element \(r\) in a ring \(R\) we write \(r + \cdots + r\) (\(n\) times) as \(nr\text{.}\) We define the characteristic of a ring \(R\) to be the least positive integer \(n\) such that \(nr = 0\) for all \(r \in R\text{.}\) If no such integer exists, then the characteristic of \(R\) is defined to be \(0\text{.}\) We will denote the characteristic of \(R\) by \(\chr R\text{.}\)

###### Example 5.17.

For every prime \(p\text{,}\) \({\mathbb Z}_p\) is a field of characteristic \(p\text{.}\) By Proposition 2.4, every nonzero element in \({\mathbb Z}_p\) has an inverse; hence, \({\mathbb Z}_p\) is a field. If \(a\) is any nonzero element in the field, then \(pa =0\text{,}\) since the order of any nonzero element in the abelian group \({\mathbb Z}_p\) is \(p\text{.}\)

###### Lemma 5.18.

Let \(R\) be a ring with identity. If \(1\) has order \(n\text{,}\) then the characteristic of \(R\) is \(n\text{.}\)

###### Proof.

If \(1\) has order \(n\text{,}\) then \(n\) is the least positive integer such that \(n 1 = 0\text{.}\) Thus, for all \(r \in R\text{,}\)

On the other hand, if no positive \(n\) exists such that \(n1 = 0\text{,}\) then the characteristic of \(R\) is zero.

###### Theorem 5.19.

The characteristic of an integral domain is either prime or zero.

###### Proof.

Let \(D\) be an integral domain and suppose that the characteristic of \(D\) is \(n\) with \(n \neq 0\text{.}\) If \(n\) is not prime, then \(n = ab\text{,}\) where \(1 \lt a \lt n\) and \(1 \lt b \lt n\text{.}\) By Lemma 5.18, we need only consider the case \(n 1 = 0\text{.}\) Since \(0 = n 1 = (ab)1 = (a1)(b1)\) and there are no zero divisors in \(D\text{,}\) either \(a1 =0\) or \(b1=0\text{.}\) Hence, the characteristic of \(D\) must be less than \(n\text{,}\) which is a contradiction. Therefore, \(n\) must be prime.

### Reading Questions Reading Questions

###### 1.

What do we mean by a zero divisor? Explain and illustrate with an example.

###### 2.

What do we mean by a unit? Which elements of the field \(\Q\) are units?

###### 3.

What is the *Cancellation Law* and how does this relate to zero divisors?

###### 4.

After reading the section, what questions do you still have? Write at least one well formulated question (even if you think you understand everything).

### Exercises Practice Problems

###### 1.

For which \(n\) is \(\Z_n\) an integral domain? For which \(n\) is \(\Z_n\) a field?

###### 2.

Just like we did for groups, we can take the direct product of rings. For example, the ring \(\Z_2 \times \Z_3\) contains 6 elements. We add and multiply elements wise.

For which \(m\) and \(n\) is the ring \(\Z_m \times \Z_n\) an integral domain?

###### 3.

Let \(R\) be a ring. For \(x \in R\text{,}\) we write \(nx\) to mean \(x + x + \cdots + x\) ( \(n\) times). What is \((nx)^2\text{?}\) What about \((nx)^3\text{?}\)

###### 4.

Suppose that \(R\) is a ring such that \(x^3 = x\) for all \(x \in R\text{.}\) Prove that \(6x = 0\) for all \(x \in R\text{.}\)

### Exercises Collected Homework

###### C1.

An element \(a\) in a ring \(R\) is called an idempotent if \(a^2 = a\text{.}\) Prove that if \(R\) is an integral domain, the only idempotents are 0 and 1. Then give an example of a ring that has an idempotent other than 0 and 1.

###### C2.

Suppose \(R\) is a ring such that \(x^2 = x\) for all \(x \in R\) (that is, every element is idempotent; such rings are called Boolean rings).

Prove that \(-x = x\) for all \(x \in R\text{.}\)

Prove that \(R\) is commutative.

For the second part, what is \((a+b)^2\text{?}\)