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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 16th 2025
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI came across this tiny page which is called by 'extension' from the page [[central extension]]. But what's this page trying to be? Merely about a certain kind of field extension? <a href="https://ncatlab.org/nlab/revision/diff/algebraic+extension/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/algebraic+extension/3">v3</a>, <a href="https://ncatlab.org/nlab/show/algebraic+extension">current</a>

   I came across this tiny page which is called by ’extension’ from the page central extension.

   But what’s this page trying to be? Merely about a certain kind of field extension?

   diff, v3, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 16th 2025
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo not a good choice given the remark at [[group extension]]: > Sometimes in the literature one sees $\hat G$ called an extension "of $A$ by $G$". This is however in conflict with terms such as _[[central extension]]_, _[[extension of principal bundles]]_, etc, where the extension is always regarded _of the base, by the [[fiber]]_. (On the other hand, our terminology conflicts with the usual meaning of "extension" in algebra. For example, in Galois theory if $k$ is a field, then an extension of $k$ contains $k$ as a subfield.)

   So not a good choice given the remark at group extension:

   Sometimes in the literature one sees $\hat G$ called an extension “of $A$ by $G$”. This is however in conflict with terms such as central extension, extension of principal bundles, etc, where the extension is always regarded of the base, by the fiber. (On the other hand, our terminology conflicts with the usual meaning of “extension” in algebra. For example, in Galois theory if $k$ is a field, then an extension of $k$ contains $k$ as a subfield.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 16th 2025
   • (edited Feb 16th 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis happens when somebody asks for a link "algebraic extension" and somebody else (often: me) decides that better than that link being broken is that it points at least to a stub entry which has at least the pointer to the Wikipedia page. Generally, the eventual purpose of any page titled "X" must be to inform readers who clicked on "X" in the hope to see more relevant information about it. (Many, many entries on the nLab do not live up to this ideal, as of now, but that's where things should be headed.)

   This happens when somebody asks for a link “algebraic extension” and somebody else (often: me) decides that better than that link being broken is that it points at least to a stub entry which has at least the pointer to the Wikipedia page.

   Generally, the eventual purpose of any page titled “X” must be to inform readers who clicked on “X” in the hope to see more relevant information about it.

   (Many, many entries on the nLab do not live up to this ideal, as of now, but that’s where things should be headed.)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 16th 2025
   • (edited Feb 16th 2025)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI was just wondering if there's a quick fix. I guess we ought to have a disambiguation page for extension which splits the concept into the $k \hookrightarrow K$ form and the $\hat{G} \to G$ form. Feels like they should have been named differently, but it is what it is. When I have a moment, I'll start something.

   I was just wondering if there’s a quick fix. I guess we ought to have a disambiguation page for extension which splits the concept into the $k \hookrightarrow K$ form and the $\hat{G} \to G$ form.

   Feels like they should have been named differently, but it is what it is. When I have a moment, I’ll start something.

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 17th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## Definition A [[field extension]] $L/K$ is __algebraic__ if every element of $L$ is a root of a nonzero [[polynomial]] with coefficients in $K$. <a href="https://ncatlab.org/nlab/revision/diff/algebraic+extension/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/algebraic+extension/4">v4</a>, <a href="https://ncatlab.org/nlab/show/algebraic+extension">current</a>

   Added:

   Definition

   A field extension $L/K$ is algebraic if every element of $L$ is a root of a nonzero polynomial with coefficients in $K$.

   diff, v4, current