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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.

DatCorno
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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.

Lee Mosher
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  • How wow. Honestly, that helps quite a lot, turns out I had the conceptual idea of modding out by x^2 + 1 all wrong. I will try to play with this a bit and internalize it somewhat. – DatCorno Nov 16 '20 at 23:30
  • @DatCorno See also this answer where i discuss how in rings with a Euclidean divison algorithm (with :smaller" remainder) we can use the remainder as normal form rep for congruence classes, which highlights the analogy between $, \Bbb Z\bmod n, $ and $,K[x]\bmod f(x),$ (or the corresponding quotient rings if you are familiar with such). – Bill Dubuque Nov 17 '20 at 09:10
  • You have both been tremendously helpful, @BillDubuque analogous reasoning of K[x] mod f(x) with the ring of integers has been eye opening. – DatCorno Nov 17 '20 at 14:33
  • @DatCorno Glad to help. You might also find of interest generalizations to multi-variate polynomial rings such as the Grobner basis algorithm, which generalizes the Euclidean division algorithm and (Gaussian) eliminations, e.g. see the introduction to Dave Bayer's thesis (under Mumford) that I posted here. – Bill Dubuque Nov 18 '20 at 00:59
  • But be sure to not focus too much on normal forms and algorithms, i.e.. it is important to alsounderstand quotient rings in general, and some of them will not have convenient normal form algorithms (indeed, in general there is no recursive algorithm for deciding equality in quotient rings). – Bill Dubuque Nov 18 '20 at 00:59
  • @BillDubuque I am planning to read Eisenbud's book on Commutative Algebra this summer so I guess I will learn about Gröbner basis there. As for your other comment, I am aware that most ideals will not be as well behaved as x^2 + 1 but it is always good to build some nice visual intuition on nicer examples before generalizing, as I am sure you will agree – DatCorno Nov 18 '20 at 03:04
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It's the same. Literally. Ideals of a ring are normal subgroups of their additive group, and the quotient of the ring by the ideal is just the quotient of the additive group by this normal subgroup. And then the multiplicative structure can be added on top. At their core, rings are groups with additional structure, and their quotients reflect this.

Vercassivelaunos
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  • I understand the formal definition of a ring and of an ideal. My grip is more about how to think about them. If think about a group, you can visualize its elements, or you can have a geometric interpretation of what it does, you can see it. But how do you see the ring of polynomials, how can you have a visual understanding of it. That is what I am struggling with. – DatCorno Nov 16 '20 at 21:42