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Let $P = \Bbb Q[x,y,z,u,v]$ and $I$ the ideal generated by: $$\begin{cases} x^5-2x^4+2x^3-x^2+1,\\ x^4+x^3y+x^2y^2+xy^3+y^4-2x^3-2x^2y-2xy^2-2y^3+2x^2+2xy+2y^2-x-y,\\ x^3+x^2y+x^2z+xy^2+xyz+xz^2+y^3+y^2z+yz^2+z^3-2x^2-2xy-2xz-2y^2-2yz-2z^2+2x+2y+2z-1,\\ x^2+xy+xz+xu+y^2+yz+yu+z^2+zu+u^2-2x-2y-2z-2u+2,\\ x+y+z+u+v-2\end{cases} $$ How can I split $P/I$ as a direct sum of minimal ideals? I solved this question using a matrix algebra that is isomorphic to $P/I$ but I want to know if there is a solution using only polynomials.

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    Why did you remove the similar question I saw yesterday? As it turns out the discriminant was a square, so your ring was a product of fields. – 2017-02-21
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    As the question was ill posed and the example did not reflect the problem I have I replaced the question as proposed by another user. – 2017-02-21
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    It’s interesting, @i.m.soloveichik, that the previous field was the real subfield of $\Bbb Q(\zeta_9)$. – 2017-02-21
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    First, you should drop $v$ and the last condition off your statement. Second, I ask whether you think that the result $\Bbb Q[x,y,z,u]/P'$ is isomorphic to a bunch of copies of $\Bbb Q[x]/(x^5-2x^4+2x^3-x^2+1$. – 2017-02-21
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    @Lubin Yes I noticed that. – 2017-02-21
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    @Bogaerts See related issues in the question: Zero-dimensional ideals in polynomial rings – 2017-02-21
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    @Lubin: Those are indeed subrings, but not necessarily ideals and certainly not likely to be direct summands, so why is this relevant? – 2017-02-22
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    My apologies for imprecise terminology. By “a bunch of fields”, I meant (and should have written) a direct sum of fields. If this is what your ring looks like, then these summands are precisely the minimal ideals. – 2017-02-22
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    @Lubin: Maybe I'm totally confused, but: Take the splitting field of a cubic that has degree $6$, There are subfields of degree $3$ generated by the individual roots, is the splitting field then a direct sum of these subfields? – 2017-02-22
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    A field is never a direct sum of proper subfields. The rings in which you want to find minimal ideals are not fields. – 2017-02-22

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