TITLE: How do you visualize polynomials rings and their quotient?
QUESTION [1 upvotes]: When working in the framework of group theory I could easily visualize the group by imagining its elements and I could see the group "collapse" when forming the quotient group by a normal subgroup by simply imagining the elements of the subgroup all being collapsed onto the identity, same thing for the other cosets.
But for polynomial rings I have a very hard thing visualizing this and it has lead to a poor understanding of ideals and quotient rings since I cannot rely on my visualize intuition anymore. I cannot make sense of $\mathbb{R}[x]$, its ideal $x^2+1$ and its quotient ring by that ideal.
I was wondering if you had any help in this regard because I am struggling here.

REPLY [3 votes]: I visualize elements of the polynomial ring $\mathbb R[x]$ as, well, polynomials, although perhaps it helps to keep one's focus on the $0$ degree term:
\begin{align*}
    & a_0 \\
a_1 x + & a_0 \\
a_2 x^2 + a_1 x + &a_0 \\
a_3 x^3 + a_2 x^2 + a_1 x + &a_0
\end{align*}
and so on.
And then, when you mod out by $x^2+1$, welp, that's just substituting every $x^2$ by $-1$, and every $x^3$ by $-x$,
\begin{align*}
    & a_0 \\
a_1 x + & a_0 \\
a_2 (-1) + a_1 x + &a_0 \\
a_3 (-x) + a_2 (-1) + a_1 x + &a_0
\end{align*}
and so on... and then you have to collect terms of course, which shifts the terms formerly known as the $x^3$ and $x^2$ terms two to the right, and shifts the terms formerly known as the $x^5$ and $x^4$ terms four to the right, and so on.
{"set_name": "stack_exchange", "score": 1, "question_id": 3910310}