Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given an ideal $I = \langle x-y,y^3+y+1 \rangle \subset \mathbb{C}[x,y]$ (this is a Gröbner basis w.r.t. degree-lexicographic order). I want to write $\mathbb{C}[x,y]/I$ as a $\mathbb{C}$-Basis and determire $\operatorname{dim}_{\mathbb{C}}\mathbb{C}[x,y]/I$. I know what a quotient ring is and how it is definied but I have no intuition how $\mathbb{C}[x,y]/I$ looks like. Any hints?

share|improve this question
Hint: $x-y\in I$ means that that in $C[x,y]/I$, $x$ and $y$ are equal. –  Thomas Andrews Aug 30 '12 at 16:34

1 Answer 1

up vote 3 down vote accepted

Well, $\mathbb{C}[x,y]/I$ is still spanned as a complex vector space by the monomials $x^i y^j$. However, that spanning set is not linearly independent: any linear combination of monomials that adds up to an element of $I$ is equal to zero!

Thinking of the Groebner basis as a rewrite scheme is useful too: the form of your basis says:

  • Whenever I see an $x$, replace it with $y$
  • Whenever I see a $y^3$, replace it with $-y-1$

which gives you an algorithm to convert any polynomial to a unique normal form... and makes it easy to see what polynomials can be normal forms.

(To be clear, whenever you see a $y^4$, that also means you see a $y^3$, because $y^4 = y^3 \cdot y$)

share|improve this answer
Thanks for the great answer! If I understand you correctly I assume all the equivalence classes are of the form $[a \cdot 1], [a \cdot y], [a \cdot y^2]$ with $a \in \mathbb{C}$. Thus, $\operatorname{dim}_{\mathbb{C}}\mathbb{C}[x,y]/I = 3$? –  joachim Aug 30 '12 at 18:11
That is indeed a basis for $\mathbb{C}[x,y]/I$. (don't forget the quotient ring also contains linear combinations of them!) –  Hurkyl Aug 30 '12 at 18:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.