Galois - User contributions [en]

MediaWiki:Sitenotice

2024-09-07T23:45:30Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/>
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-09-07T23:43:26Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>\sqrt{7 + \sqrt{4}}!! + 4! = 744</math>
* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>\sqrt{7 - 4}!! + 4! = 744</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>(3 + 4)^3 = 343</math>

MediaWiki:Sitenotice

2024-09-06T17:20:44Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.'''<br/>
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/>
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

Galois:Enabling site search autocompletion

2024-08-07T21:30:14Z

Vipul: /* How to fix it */

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Galois).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

Galois:429 Too Many Requests error

2024-08-07T20:36:47Z

Vipul:

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

Galois:429 Too Many Requests error

2024-08-07T20:35:30Z

Vipul:

This content is copied from [[Ref:Ref:429 Too Many Request error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

Galois:429 Too Many Requests error

2024-08-07T04:25:39Z

Vipul: Vipul moved page Galois:429 Too Many Request error to Galois:429 Too Many Requests error

This content is copied from [[Ref:Ref:429 Too Many Request error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you may need to send both; the server uses IPv6 if your client has both addresses. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

MediaWiki:Sitenotice

2024-08-07T04:25:18Z

Vipul:

Galois:429 Too Many Requests error

2024-08-07T04:24:51Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Request error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual hu..."

This content is copied from [[Ref:Ref:429 Too Many Request error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you may need to send both; the server uses IPv6 if your client has both addresses. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

Galois:Enabling site search autocompletion

2024-08-07T04:18:33Z

Vipul:

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Galois).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site, and go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

File:Site search autocompletion working.png

2024-08-07T04:18:04Z

Vipul:

File:Site search autocompletion broken.png

2024-08-07T04:17:28Z

Vipul:

Galois:Enabling site search autocompletion

2024-08-07T04:14:20Z

Vipul: Created page with "Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below: File:Site search autocompletion broken.png Note that the actual search is still working -- y..."

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site, and go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

User:Vipul/Sandbox

2024-08-03T23:46:36Z

Vipul:

User:Vipul/Sandbox

2024-08-03T23:41:58Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>\sqrt{7 + \sqrt{4}}!! + 4! = 744</math>
* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>\sqrt{7 - 4}!! + 4! = 744</math>
* <math>2^{8 - 1} = 128</math>

User:Vipul/Sandbox

2024-08-03T23:04:28Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>\sqrt{7 + \sqrt{4}}!! + 4! = 744</math>
* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>\sqrt{7 - 4}!! + 4! = 744</math>

Field with trivial Brauer group

2024-07-20T05:35:06Z

Vipul:

{{field property}}

==Definition==

A '''field with trivial Brauer group''' is a [[field]] whose [[defining ingredient::Brauer group]] is the [[trivial group]].

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Quasi-algebraically closed field]]
** [[Weaker than::Algebraically closed field]]
** [[Weaker than::Finite field]]
** [[Weaker than::Separably closed field]]

User:Vipul/Sandbox

2024-07-14T21:24:27Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>\sqrt{7 + \sqrt{4}}!! + 4! = 744</math>
* <math>9^{\sqrt{7 + 2}} = 729</math>

User:Vipul/Sandbox

2024-07-14T21:16:04Z

Vipul:

* <math>\sqrt{7 + 2}!! + 3 = 723</math>
* <math>\sqrt{7 + \sqrt{4}}!! + 4! = 744</math>

User:Vipul/Sandbox

2024-07-14T21:13:04Z

Vipul: Created page with "* <math>\sqrt{7 + 2}!! + 3 = 723</math>"

* <math>\sqrt{7 + 2}!! + 3 = 723</math>

Template:Top notice

2024-07-14T21:12:05Z

Vipul:

{{quotation|Welcome to '''{{fullsitetitle}}'''. This is a pre-alpha stage field theory-cum-Galois theory wiki primarily managed by [[User:Vipul|Vipul Naik]], a Ph.D. in Mathematics at the University of Chicago. It is part of a broader subject wikis initiative -- see the [[Ref:Main Page|subject wikis reference guide]] for more details.}}

MediaWiki:Sitenotice

2024-07-14T21:11:43Z

Vipul: Blanked the page

MediaWiki:Sitenotice

2024-07-14T21:11:06Z

Vipul:

[[Main Page|{{fullsitetitle}} ({{sitestatus}})]]

Hilbert's Theorem 90

2012-01-07T20:38:41Z

Vipul: Created page with "==Statement== ===In terms of group cohomology=== Suppose <math>L/K</math> is a (not necessarily finite) fact about::Galois extension with [[Galois grou..."

==Statement==

===In terms of group cohomology===

Suppose <math>L/K</math> is a (not necessarily [[finite extension|finite]]) [[fact about::Galois extension]] with [[Galois group]] <math>G</math>. <math>G</math> has a natural action on the [[multiplicative group of a field|multiplicative group]] <math>L^\times</math>. The theorem states that the [[first cohomology group]] for this group action is the [[trivial group]].

===Explicit statement for cyclic extensions===

{{fillin}}

Normal basis theorem

2012-01-07T20:34:12Z

Vipul: Created page with "==Statement== Suppose <math>L/K</math> is a finite Galois extension of fields with Galois group <math>G</math>. Then, there exists an element <math>\alpha \in L</mat..."

==Statement==

Suppose <math>L/K</math> is a [[finite Galois extension]] of [[field]]s with [[Galois group]] <math>G</math>. Then, there exists an element <math>\alpha \in L</math> such that the set:

<math>\! \{ \sigma(\alpha) \mid \sigma \in G \}</math>

forms a basis for <math>L</math> as a vector space over <math>K</math>. In other words, we can always find a basis that is a single orbit under the action of the Galois group. Such a basis is termed a ''normal basis''.

Primitive element theorem

2012-01-07T20:32:07Z

Vipul: /* Related facts */

==Statement==

Suppose <math>L/K</math> is a [[fact about::finite extension]] of fields, i.e., the [[degree of a field extension|degree]] of the extension is finite. The '''primitive element theorem''' states that there exists an element <math>\alpha \in L</math> such that <math>L = K(\alpha)</math>. In other words, the extension can be generated by a ''single'' element over <math>K</math>. Such an element is termed a ''primitive element''.

==Related facts==

* [[Normal basis theorem]] states that any [[finite Galois extension]] has a basis (as a vector space over the base field) that forms a single orbit under the action of the [[Galois group]].

Primitive element theorem

2012-01-07T20:28:09Z

Vipul: Created page with "==Statement== Suppose <math>L/K</math> is a fact about::finite extension of fields, i.e., the degree of the extension is finite. The '''primi..."

==Statement==

Suppose <math>L/K</math> is a [[fact about::finite extension]] of fields, i.e., the [[degree of a field extension|degree]] of the extension is finite. The '''primitive element theorem''' states that there exists an element <math>\alpha \in L</math> such that <math>L = K(\alpha)</math>. In other words, the extension can be generated by a ''single'' element over <math>K</math>. Such an element is termed a ''primitive element''.

==Related facts==

* [[Normal basis theorem]] states that any [[Galois extension]] has a basis (as a vector space over the base field) that forms a single orbit under the action of the [[Galois group]].

Separable polynomial

2009-05-15T23:29:08Z

Vipul:

{{univariate polynomial upto associates property}}

==Definition==

Let <math>K</math> be a field and <math>f(x) \in K[x]</math> be a nonzero polynomial. We say that <math>f</math> is a '''separable polynomial''' if the following equivalent conditions are satisfied:

# <math>f(x)</math> and its [[defining ingredient::formal derivative]] <math>f'(x)</math> are relatively prime in <math>K[x]</math>.
# <math>f(x)</math> splits completely into ''distinct'' linear factors over its splitting field.
# For any field <math>L</math> containing <math>K</math>, <math>f</math> is a square-free polynomial over <math>L</math>, i.e., no square of a polynomial divides <math>f</math>.
# The [[defining ingredient::discriminant of a polynomial|discriminant]] of <math>f</math> is nonzero.

Galois:Copyrights

2009-05-15T02:00:01Z

Vipul: Created page with 'This is a common copyright notice to all subject wikis. Original notice available at Ref:Ref:Copyrights. ==General license information== All content is put up under the [ht...'

This is a common copyright notice to all subject wikis. Original notice available at [[Ref:Ref:Copyrights]].

==General license information==

All content is put up under the [http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribute Share-Alike 3.0 license (unported)] (shorthand: CC-by-SA). Read the [http://creativehttp://ref.subwiki.org/w/skins/common/images/button_media.pngcommons.org/licenses/by-sa/3.0/legalcode full legal code here].

===Exceptions===

The software engine that runs the wiki is MediaWiki, and this software is licensed under the GNU General Public License, which is distinct from and currently incompatible with the Creative Commons license. Similarly, some of the extensions to the software engine are licensed by their developers using the GPL, which is incompatible with Creative Commons licenses.

More information on the version of MediaWiki and extensions: [[Special:Version]].

In addition, some templates may have been copied or adapted from Wikipedia or other wiki-based websites that use licenses other than Creative Commons licenses. Wikipedia uses the GNU Free Documentation License (GFDL) that, as of now, is incompatible with CC licenses. In such cases, it is clearly indicated on the page that it is licensed differently.

===Short description of the license===

The CC-by-SA license has the following features:

# Content can be used and reused for any purposes, commercial or noncommercial.
# Any public use or reuse of the material requires attribution of the original source (for attribution formats, see later in the document).
# In case of adaptation or creation of derivative works, the newly created works must also be licensed under the same license (CC-by-SA) unless the creator of the derivative work takes ''explicit'' permission from the creator of the original work.

===Fair use and other rights===

The license cannot and does not limit fair use rights. Fair use may include quoting or copying select passages for the purposes of criticism and review.

The license also cannot limit the use of data or information contained in the content on subject wikis. This means that anybody can use or reuse the facts or knowledge they gain from the subject wiki in any way they want.

==Applicability of license to the subject wiki==

===Examples of personal use that do not necessitate use of the license terms===

* Saving, making copies, or taking printouts of pages on the subject wiki, as long as these copies are meant for personal use and are not to be distributed to others.
* Learning facts from the subject wikis and using those facts as part of instruction or research.
* Providing links to pages on the subject wikis.
* Quoting short passages from the subject wiki in a criticism or review (it is preferable, though not legally enforceable, that a link to the original content being reviewed be provided).

===Examples of personal use that necessitate use of the license terms===

* Creating a mirror website by exporting pages or content in bulk: In this case, the mirror website must release the mirrored content under the same license terms and ''also'' attribute the original source with a link. Attribution formats are discussed below.
* Giving handouts of pages or collections of pages to a group of people, such as students in a course: In this case, the handouts should be accompanied by an attribution that makes clear both the source and the original license terms, as well as a link to the Uniform Resource Identifier (URI) of the subject wiki.

Attribution formats:

* For websites: ''This content is from [{{fullurl:Main Page}} {{SITENAME}}]'' or ''This content is from [{{fullurl:Main Page}} {{fullsitetitle}}]''. Preferably, also a link to the specific pages (if any) used. The language of the attribution may be modified somewhat to indicate whether the content has been used as such, or whether changes have been made.
* For print materials: ''This content is from {{SITENAME}}. The site URL is http://{{SITENAME}}.subwiki.org''

Attribution of individuals who have contributed content to the subject wiki entries is ''not'' necessary. (However, this list of individuals can be tracked down, as discussed in the next section).

==Uses not within the ambit of the license or fair use rights==

Those who have contributed all the content to a particular subject wiki entry can republish the same content anywhere else ''without restriction'' and can also release the content elsewhere under a different license, since they own the full copyright to their content.

To use the content on the wiki in a manner that is outside the terms of the license or beyond fair use rights, the individuals who have contributed that particular content need to be contacted. For any particular page, this information can be accessed using the history tab of the page. The users who have made edits to the page are the ones who need to be contacted.

In case of difficulty doing this, please contact vipul.wikis@gmail.com or vipul@math.uchicago.edu for more information.

==Copyright violations==

All users using subject wikis are requested not to violate copyright of other authors or publishers. This includes providing links that may aid and abet copyright violation. Examples of things that may be considered copyright violation:

* Putting up links to personal copies of restricted-access journal articles acquired through a personal or library subscription: This usually goes against the Terms of Service of the subscription. The exception is when the content of the articles is out of copyright.
* Giving links to personal copies, or copies on filesharing systems, of books that are under copyright and where either owning or sharing a copy in that manner is against the terms of copyright.

In case copyright violations are detected, please email vipul.wikis@gmail.com and vipul@math.uchicago.edu to have the matter looked into immediately.

Discriminant of a polynomial

2009-05-15T01:59:20Z

Vipul: Created page with '==Definition== Suppose <math>K</math> is a field and <math>f(x) \in K[x]</math> is a nonconstant polynomial. The '''discriminant''' of <math>f</math> is defined in the follo...'

==Definition==

Suppose <math>K</math> is a [[field]] and <math>f(x) \in K[x]</math> is a nonconstant polynomial. The '''discriminant''' of <math>f</math> is defined in the following equivalent ways:

# It is the [[defining ingredient::resultant polynomial]] of <math>f(x)</math> and its [[defining ingredient::formal derivative of a polynomial|formal derivative]] <math>f'(x)</math>.
# Let <math>L</math> be a splitting field for <math>f</math> over <math>K</math>. Over this, write:

<math>f(x) = a^n \prod_{i=1}^n (x - \alpha_i)</math>.

Then, the discriminant of <math>f</math> is defined as:

<math>a^n \prod_{1 \le i < j \le n} (\alpha_i - \alpha_j)^2</math>.

The discriminant of a linear polynomial <math>ax + b</math> is defined to be <math>a</math>, because the product in this case is empty.

The discriminant of a polynomial is nonzero if and only if the polynomial is a [[separable polynomial]].

Formal derivative of a polynomial

2009-05-15T01:52:39Z

Vipul: /* Facts */

==Definition==

Suppose <math>K</math> is a [[field]] and <math>f(x) \in K[x]</math> is a polynomial. The '''formal derivative''' of <math>f</math>, denoted <math>f'</math>, is defined as follows.

If <math>f(x) = \sum_{m=0}^n a_mx^m</math>, then <math>f'(x) = \sum_{m=1}^n ma_mx^{m-1}</math>.

Here, the <math>ma_m</math> is understood as <math>a_m</math> added to itself <math>m</math> times.

==Facts==

The formal derivative gives a <math>K</math>-linear map from <math>K[x]</math> to itself. When <math>K</math> has characteristic zero, the kernel of the map is <math>K</math>, whereas when <math>K</math> has characteristic <math>p</math>, the kernel of the map is <math>K[x^p]</math>.

The formal derivative also satisfies the following rule for multiplication, called the Leibniz rule:

<math>\! (fg)'(x) = f'(x)g(x) + f(x)g'(x)</math>.

Formal derivative of a polynomial

2009-05-15T01:52:05Z

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Formal derivative

2009-05-15T01:48:44Z

Vipul: Redirected page to Formal derivative of a polynomial

#redirect [[Formal derivative of a polynomial]]

Artin-Schreier extension

2009-05-15T01:44:56Z

Vipul: Created page with '==Definition== Let <math>K</math> be a field of characteristic <math>p</math>, and let <math>a \in K</math> be such that the polynomial <math>x^p - x - a</math> is irreducible o...'

==Definition==

Let <math>K</math> be a field of characteristic <math>p</math>, and let <math>a \in K</math> be such that the polynomial <math>x^p - x - a</math> is irreducible over <math>K</math>. The '''Artin-Schreier extension''' corresponding to <math>a</math> is the field

<math>K[x]/(x^p - x - a)</math>

which is ''also'' equal to the splitting field of <math>x^p - x - a</math> over <math>K</math>. It is a [[Galois extension]] whose Galois group is a cyclic group of order <math>p</math>.

In fact, any Galois extension of degree <math>p</math> over a field of characteristic <math>p</math> can be realized as an Artin-Schreier extension.

==Facts in the definition==

* [[Artin-Schreier polynomial is either irreducible or splits into linear factors]]

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Cyclic extension]]
* [[Stronger than::Galois extension]]

Cyclotomic polynomial

2009-05-15T01:34:59Z

Vipul: Created page with '==Definition== Let <math>n</math> be a natural number. The '''cyclotomic polynomial''' of degree <math>n</math>, denoted <math>\Phi_n</math>, is defined in the following way...'

==Definition==

Let <math>n</math> be a [[natural number]]. The '''cyclotomic polynomial''' of degree <math>n</math>, denoted <math>\Phi_n</math>, is defined in the following ways:

* It is the product, over all primitive <math>n^{th}</math> roots <math>\zeta</math> of <math>1</math>, of the linear polynomials <math>x - \zeta</math>.
* It is the [[minimal polynomial]] of any primitive <math>n^{th}</math> root of unity.

===Equivalence of definitions===

The equivalence of definitions essentially follows by using definition (1) and showing, from that definition, that [[cyclotomic polynomials are irreducible]].

Cyclotomic extension

2009-05-15T01:32:16Z

Vipul: Created page with '{{field extension property}} ==Definition== A field extension is termed a '''cyclotomic extension''' if it is obtained as the splitting field of polynomials of the form <ma...'

{{field extension property}}

==Definition==

A [[field extension]] is termed a '''cyclotomic extension''' if it is obtained as the splitting field of polynomials of the form <math>x^n - 1</math>, where <math>n</math> is not a multiple of the characteristic of the field. In other words, it is an extension obtained by adjoining primitive <math>n^{th}</math> roots of unity.

Kronecker-Weber theorem

2009-05-15T01:29:51Z

Vipul: Created page with '==Statement== Any finite fact about::abelian extension of the field of rational numbers is contained in a [[fact about::cyclotomic exten...'

==Statement==

Any [[fact about::finite extension|finite]] [[fact about::abelian extension]] of the [[field of rational numbers]] is contained in a [[fact about::cyclotomic extension]] of the field of rational numbers.

Galois group

2009-05-15T01:28:50Z

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==Definition==

===For a field extension===

The '''Galois group''' of a [[field extension]] is defined as the [[automorphism group of a field extension|automorphism group of the field extension]].

===For a polynomial===

The '''Galois group''' of a [[separable polynomial]] (or more generally, a polynomial all of whose irreducible factors are separable) is defined as the Galois group of the [[splitting field]] of that polynomial.

Abelian extension

2009-05-15T01:27:22Z

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{{field extension property}}

==Definition==

An '''abelian extension''' is a [[Galois extension]] whose [[Galois group]] is an [[abelian group]].

Global field

2009-05-15T01:17:01Z

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{{field property}}

==Definition==

A '''global field''' is a [[field]] that is either a [[defining ingredient::number field]] or the [[defining ingredient::function field]] of an algebraic curve over a [[finite field]].

Separable polynomial

2009-05-15T01:15:39Z

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Radical extension

2009-05-14T23:31:04Z

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{{field property}}

==Definition==

===For finite extensions===

Suppose <math>L</math> is a [[finite extension]] of a [[field]] <math>K</math>. We say that <math>L</math> is a '''radical extension''' if there exist intermediate fields <math>K = K_0 \subseteq K_1 \subseteq \dots \subseteq K_r = L</math> such that each <math>K_i</math> is a [[defining ingredient::simple radical extension]] of <math>K_{i-1}</math>: In other words, each <math>K_i</math> is obtained by adjoining to <math>K_{i-1}</math> a root of a polynomial of the form <math>x^n - a</math>.

Cyclic extension

2009-05-14T23:27:56Z

Vipul: Created page with '{{field extension property}} ==Definition== A '''cyclic extension''' is a finite Galois extension of fields whose Galois group is a cyclic group.'

{{field extension property}}

==Definition==

A '''cyclic extension''' is a finite [[Galois extension]] of fields whose [[Galois group]] is a [[cyclic group]].

Totally real number field

2009-05-14T23:23:54Z

Vipul: Created page with '{{field property}} ==Definition== A '''totally real number field''' is a number field such that any subfield of the field of complex numbers isomorphic to it is con...'

{{field property}}

==Definition==

A '''totally real number field''' is a [[number field]] such that any [[subfield]] of the [[field of complex numbers]] isomorphic to it is contained in the [[field of real numbers]].

Fundamental theorem of algebra

2009-05-14T23:22:24Z

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==Statement==

The [[field of complex numbers]] is an [[algebraically closed field]].

Absolute Galois group

2009-05-14T23:21:07Z

Vipul: Created page with '==Definition== The '''absolute Galois group''' of a field <math>K</math> is defined as the group <math>\operatorname{Aut}(\overline{K}/K)</math> where <math>\overline{K}</ma...'

==Definition==

The '''absolute Galois group''' of a [[field]] <math>K</math> is defined as the group <math>\operatorname{Aut}(\overline{K}/K)</math> where <math>\overline{K}</math> denotes the [[separable closure]] of <math>K</math>.

The group is not just an abstract group but comes with the structure of a [[profinite group]]: it is the inverse limit of the automorphism groups for all finite Galois extensions. This also endows it with a topology (the Krull topology).

Purely inseparable extension

2009-05-14T23:16:24Z

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{{field extension property}}

==Definition==

Suppose <math>L</math> is a [[field extension]] of a [[field]] <math>K</math>. We say that <math>L</math> is '''purely inseparable''' over <math>K</math> if it satisfies the following equivalent conditions:

# The extension is a [[defining ingredient::normal extension]] and its [[defining ingredient::field extension with trivial automorphism group|automorphism group is trivial]].
# <math>L</math> is an [[algebraic extension]] of <math>K</math> such that if <math>M</math> is a [[normal extension]] of <math>K</math> containing <math>L</math>, the fixed field of <math>\operatorname{Aut}(M/K)</math> contains <math>L</math>.
# The [[minimal polynomial]] of any element of <math>L</math> over <math>K</math> is a power of a linear polynomial.
# If <math>K</math> has characteristic zero, <math>K = L</math>. If <math>K</math> has characteristic <math>p</math>, every element <math>a \in L</math> satisfies <math>a^{p^n} \in K</math> for some <math>n</math> (depending on <math>a</math>).

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Normal extension]]
* [[Stronger than::Algebraic extension]]

===Opposite properties===

* [[Galois extension]] is a normal extension that is also a [[separable extension]].

Castelnuovo's theorem

2009-05-14T23:09:11Z

Vipul:

==Statement==

Any sub-extension of a [[fact about::purely transcendental extension]] of transcendence degree <math>2</math> is also a purely transcendental extension.

==Related facts==

* [[Lüroth's theorem]]

Castelnuovo's theorem

2009-05-14T23:08:35Z

Vipul: Created page with '==Statement== Any sub-extension of a purely transcendental extension of transcendence degree <math>2</math> is also a purely transcendental extension.'

==Statement==

Any sub-extension of a [[purely transcendental extension]] of transcendence degree <math>2</math> is also a purely transcendental extension.

Lüroth's theorem

2009-05-14T23:07:52Z

Vipul:

==Statement==

Let <math>K</math> be a field, and <math>K(t)</math> be the [[field of univariate rational functions]], i.e., the field of rational functions in one variable over <math>K</math>. Then, any subfield of <math>K(t)</math> ''properly'' containing <math>K</math> is of the form <math>K(f(t))</math>, where <math>f</math> is a rational function. In particular, this subfield is again the field of univariate rational functions, with the variable now being <math>f(t)</math>.

In other words, any nontrivial sub-extension of a simple transcendental extension is again a simple transcendental extension.

==Related facts==

* [[Castelnuovo's theorem]]