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In mathematics a '''Steinberg symbol''' is a pairing function which generalises the [[Hilbert symbol]] and plays a role in the [[algebraic K-theory]] of [[field (mathematics)|fields]].  It is named after mathematician [[Robert Steinberg]].
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


For a field ''F'' we define a ''Steinberg symbol'' (or simply a ''symbol'') to be a function
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
<math>( \cdot , \cdot ) : F^* \times F^* \rightarrow G</math>, where ''G'' is an abelian group, written multiplicatively, such that
* Only registered users will be able to execute this rendering mode.
* <math>( \cdot , \cdot ) </math> is bimultiplicative;
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.
* if <math>a+b = 1</math> then <math>(a,b) = 1</math>.


The symbols on ''F'' derive from a "universal" symbol, which may be regarded as taking values in <math>F^* \otimes F^* / \langle a \otimes 1-a \rangle</math>.  By a theorem of Matsumoto, this group is <math>K_2 F</math> and is part of the [[Milnor K-theory]] for a field.
Registered users will be able to choose between the following three rendering modes:


==Properties==
'''MathML'''
If (⋅,⋅) is a symbol then (assuming all terms are defined)
:<math forcemathmode="mathml">E=mc^2</math>
* <math> (a, -a) = 1 </math>;
* <math> (b, a) = (a, b)^{-1} </math>;
* <math> (a, a) = (a, -1) </math> is an element of order 1 or 2;
* <math> (a, b) = (a+b, -b/a) </math>.


==Examples==
<!--'''PNG''' (currently default in production)
* The [[Hilbert symbol]] on ''F'' with values in {±1} defined by<ref>{{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=A Course in Arithmetic | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | volume=7 | isbn=978-3-540-90040-5 | year=1996 }}</ref>
:<math forcemathmode="png">E=mc^2</math>
:<math>(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\in F^3;\\-1,&\mbox{ if  not.}\end{cases}</math>


==See also==
'''source'''
* [[Steinberg group (K-theory)]]
:<math forcemathmode="source">E=mc^2</math> -->


==References==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
{{reflist}}
* {{cite book | first1=P.E. | last1=Conner | first2=R. | last2=Perlis | title=A Survey of Trace Forms of Algebraic Number Fields | series=Series in Pure Mathematics | volume=2 | publisher=World Scientific | year=1984 | isbn=9971-966-05-0 | zbl=0551.10017 }}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | pages=132–142 }}
* {{cite journal | first=Robert | last=Steinberg | authorlink=Robert Steinberg | title=Générateurs, relations et revêtements de groupes algébriques | journal=Colloq. Théorie des Groupes Algébriques | location=Bruxelles | year=1962 | publisher=Gauthier-Villars | pages=113–127 | mr=MR0153677 | zbl=0272.20036 | language=French }}


==External links==
==Demos==
*[http://eom.springer.de/S/s130540.htm Steinberg symbol] at the [[Encyclopaedia of Mathematics]]


[[Category:K-theory]]
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


{{algebra-stub}}
 
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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