This is not very intuitive:

sage: sqrt(3) in QQ[sqrt(3)]
False
sage: QQ[sqrt(3)]
Number Field in sqrt3 with defining polynomial x^2 - 3


On Sun, Oct 07, 2012 at 08:23:20AM -0700, Greg Laun wrote:
> __contains__ is broken in general and special linear groups. For example:
> 
> sage: identity_matrix(GF(2),2) in GL(2,CC)
> True
> 
> The base field of the matrix is characteristic 2, but it's reported as being 
> a matrix over C.
> 
> I reported this to sage-devel here: 
> https://groups.google.com/forum/?fromgroups=#!searchin/sage-devel/laun/sage-devel/WwovHNaUldE/lt2pFK6BeJsJ
> It contains some discussion, including why using the boilerplate __contains__ 
> probably won't work.
> 
> I was about to submit a bug report, when I noticed the wiki telling me to 
> email sage-support first.  I already emailed sage-devel, so I hope I'm not 
> spamming too much by also emailing here.
> 
> 

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