Degree of a field extension

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Revision 1078

8/8/2026, 9:33:28 AM · Ancient Tree

Updated linked exercises

linked exercises~~Field extension of $\R$ of odd degree~~Field extension of

\R

of odd degree, Field extensions of

\mathbb{C}

of degree 2

Revision 1058

8/7/2026, 8:23:06 PM · Ancient Tree

Updated linked exercises

linked exercises~~None~~Field extension of

\R

of odd degree

Revision 1056

8/7/2026, 7:35:45 PM · Ancient Tree

Concept created

If $E$ is a [[Field extension|field extension]] of a field $K$, then $E$ can be viewed as a [[Vector space|vector space]] over $K$.

The degree of the extension $K\subseteq E$, denoted as $[E:K]$, is the [[dimension of a vector space|dimension]] of $K$ viewed as a vector space over $K$.

##### Examples
- $\R\subseteq \mathbb{C}$ has degree 2, since $\{1,i\}$ forms a basis of $\mathbb{C}$ as a vector space over $\R$.