§16 Fields

§16.1 Basic Concepts of Fields

Definition 16.1 A commutative ring is called a field if

is a group under multiplication.

Example 16.1 $\mathbb{R}$, $\mathbb{Q}$, and $\mathbb{C}$ are fields because every nonzero element has a multiplicative inverse.

Example 16.2 $\mathbb{Z}$ is not a field because any integer other than $\pm1$ has no multiplicative inverse.

§16.2 Subfields

§16.3 Prime Fields

§16.4 Characteristic of a Field

§16.5 Field Extensions