Exercise 569Algebra › Rings and fieldsL1Added 3 Nov 2026

Fields

#field#examples#invertible elements

Statement

Which of the following are fields?

(a) $\mathbb{Q}$ ; (b) $\mathbb{R}$ ; (c) $\mathbb{C}$ ; (d) $\mathbb{Z}$ ; (e) $\mathbb{Z}/5\mathbb{Z}$ ; (f) $\mathbb{Z}/4\mathbb{Z}$ .

Hint

A field is a commutative ring with unit where every nonzero element is invertible.

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(a),(b),(c) $\mathbb{Q},\mathbb{R},\mathbb{C}$ are fields: every nonzero element has an inverse in the same set.

(d) $\mathbb{Z}$ : not a field ( $2$ has no inverse in $\mathbb{Z}$ ).

(e) $\mathbb{Z}/5\mathbb{Z}$ : $5$ prime, so $\mathbb{Z}/5\mathbb{Z}$ is a field ( $\mathbb{F}_5$ ).

(f) $\mathbb{Z}/4\mathbb{Z}$ : $\bar2$ is not invertible ( $\bar2\cdot\bar k\in\{\bar0,\bar2\}$ ), so not a field.

Conclusion. $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{Z}/5\mathbb{Z}$ are fields; $\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ are not.