Basis of quotient ring as a vector space

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dd0dd0
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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 6 years ago

rburing

10964 ●4 ●80 ●219 https://www.rburing.nl/

 I.normal_basis()

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Thanks! This works perfectly

dd0dd0 ( 6 years ago )

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 ( 6 years ago )

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