Definition:Field Extension/Degree

Definition

Let $E / F$ be a field extension of a field $F$.

The degree of $E / F$, denoted $\index E F$, is the dimension of $E / F$ when $E$ is viewed as a vector space over $F$.

Finite

$E / F$ is a finite field extension if and only if its degree $\index E F$ is finite.

Infinite

$E / F$ is an infinite field extension if and only if its degree $\index E F$ is not finite.

Also see

• Definition:Finite Field Extension: where $\index E F$ is finite
• Definition:Infinite Field Extension: where $\index E F$ is not finite
• Definition:Separable Degree
• Definition:Inseparable Degree

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 36$. The Degree of a Field Extension