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What are elements in the quotient ring $\mathbb{R}[x,y] / (x^2 + y^2 +1)$?

In the case of $\mathbb{R} / (f(x))$, I can visualize the elements by using the division algorithm. But since in $\mathbb{R}[x,y]$, the division algorithm is not available, how do I know what are the elements in this quotient ring $\mathbb{R}[x,y] / (x^2 + y^2 +1)$?

Andy
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  • In the case of $\mathbb{R} / (f(x))$ there is no non-constant polynomial to divide. For the other ring, it is just the quotient ring by the ideal generated by $x^2+y^2+1$, so residue classes of polynomials $f(x,y)$. For more details see the posts here, e.g., this one. – Dietrich Burde Feb 14 '22 at 19:35
  • @DietrichBurde I'm looking for an explicit description of the elements in this quotient ring. If that is not possible, I want to at least get an idea of what are some properties of this quotient ring. – Andy Feb 14 '22 at 19:50
  • For the properties see the posts here (see the comment above). What do you mean by "explicit description"? Every element is explicitly a residue class, i.e., $f(x,y)\mod I$, where $I=(x^2+y^2+1)$. I think what you expect just is not possible. – Dietrich Burde Feb 14 '22 at 19:52
  • @DietrichBurde I know $\mathbb{R}[x] / ( x^2 +1 )$ can be written as $ { ax + b \mid a, b \in \mathbb{R} }$, is there a analogy in $\mathbb{R}[X,Y]/ (x^2 + y^2 + 1)$? – Andy Feb 14 '22 at 19:59

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In that ring, $y^2 = -1- x^2$, and so, powers of $y$ with exponent greater than $1$ can be eliminated in the expression of residue classes. That is, every $p(x,y)\in \mathbb{R}[x,y]/(x^2+y^2+1)$ can be written:

$$p(x,y) = a_{0}(y) + a_1(y)x + a_n(y)x^2 + \dots + a_n(y)x^n$$

where each $a_i(y) = b_i + c_iy$ is a polynomial of degree one.

Another way to understand this ring is as the ring of real polynomial functions defined on the subvariety given by the equation:

$$x^2 + y^2 = -1$$

It has no real points, so it's more natural to consider it as a complex variety. Still, you can consider polynomials with real coefficients restricted to it, and that gives the ring $\mathbb{R}[x,y]/(x^2+y^2+1)$.

Compacto
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