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Basis of quotient ring as a vector space

asked 2019-02-03 12:59:06 -0500

dd0dd0 gravatar image

I'm curious how I might get a k-basis for a quotient ring of the form R = k[x_1,...x_n] / I, where I is some ideal in the polynomial ring.

For example, given I = {x^3}, and R = Q[x]/I, I would want Sage to give me the vector space basis {1,x,x^2}.

Any help is appreciated!! :)

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answered 2019-02-03 13:48:07 -0500

rburing gravatar image
 I.normal_basis()
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Thanks! This works perfectly

dd0dd0 gravatar imagedd0dd0 ( 2019-02-03 14:28:51 -0500 )edit
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You're welcome! If you want to know the algorithm behind it, I think it is done by using a Groebner basis $I = \langle g_1, \ldots, g_s \rangle$: take all the monomials which are not in $LM(I) = \langle LM(g_1), \ldots, LM(g_s) \rangle$ (at least this is a way to do it).

rburing gravatar imagerburing ( 2019-02-03 15:19:08 -0500 )edit

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Asked: 2019-02-03 12:59:06 -0500

Seen: 56 times

Last updated: Feb 03