Ivan Shishkin, Birch Grove

Field extension

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Revision 823

8/1/2026, 7:20:12 PM · Sequoia

Concept edited

Recorded titleField extension
Recorded typeDefinition
Compare with revision 3102 changed lines
1A field extension is a [[homomorphism of fields|homomorphism of fields]] $K\rightarrow L$. It is denoted as $K\subseteq L$.
1A field extension is a [[homomorphism of fields|homomorphism of fields]] $K\hookrightarrow L$. It is denoted as $K\subseteq L$.
2
3##### Remarks
4- Any homomorphism of fields is injective, so $K$ can be identified with its [[Image of a map|image]] in $L$; we simply regard $K$ as a [[Subfield|subfield]] of $L$.

Revision 310

7/7/2026, 3:03:27 PM · Ancient Tree

Concept edited

This older revision predates detailed metadata tracking.

Revision 301

7/7/2026, 2:49:32 PM · Ancient Tree

Concept edited

Compare with revision 2982 changed lines
1A field extension is a [[homomorphism of fields|homomorphism of fields]] $K\rightarrow L$. It is denoted as $K\subseteq L$.
2
3##### Remarks
4- Any homomorphism of fields is injective, so $K$ can be identified with its image in $L$; we simply regard $K$ as a subfield of $L$.
4- Any homomorphism of fields is injective, so $K$ can be identified with its [[Image of a map|image]] in $L$; we simply regard $K$ as a [[Subfield|subfield]] of $L$.

Revision 298

7/7/2026, 2:46:54 PM · Ancient Tree

Concept created

A field extension is a [[homomorphism of fields|homomorphism of fields]] $K\rightarrow L$. It is denoted as $K\subseteq L$.

##### Remarks 
- Any homomorphism of fields is injective, so $K$ can be identified with its image in $L$; we simply regard $K$ as a subfield of $L$.