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The ring of infinitesimals is the quotient ring $\mathbb{R}[x] / x^2\mathbb{R}[x].$ What is $(3+x)(3-x)$ in this ring?

I don't quite understand what this question is asking about.

My thought is that

since $(3+x)(3-x) = 9 - x^2$, it's in the coset of $9 + x^2\mathbb{R}[x]$. When the problem asks that "What is $(3+x)(3-x)$ in this ring", is it asking which coset $(3+x)(3-x)$ is in? So just $9$?

jun
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    So what don't you quite understand then? The questions asks for an element $f$ in this ring $R$ with $f=(3+x)(3-x)$, not as a product, but as the class of a polynomial. – Dietrich Burde Oct 19 '20 at 13:30
  • oh so it's asking what the $f$ in $\mathbb{R}[x] $ transforms into in the given quotient ring? – jun Oct 19 '20 at 13:39
  • It transforms into the constant polynomial $9$, or the class of $9$. Usually we just write $9$. – Dietrich Burde Oct 19 '20 at 13:50
  • It's analogous to asking: what is $3\cdot 4$ in $\Bbb Z_{10}$ or $\bmod 10$?. It refers to the canonical image in the quotient ring $\ a \mapsto [a],,$ mapping an element to its coset in the quotient. Here the image is the first order taylor approximation ("tangent"). This ring is known as the algebra of dual numbers. It provides a convenient algebraic model of tangent and jet spaces. – Bill Dubuque Oct 19 '20 at 21:18

1 Answers1

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The quotient ring is the set of modulo-$x^2$ equivalence classes of elements of $\Bbb R[x]$. The unique polynomial equivalent to $9-x^2$ of degree less than that of $x^2$ is a degree-$0$ polynomial, $x\mapsto9$. So whether the question asks for this polynomial or the equivalence class containing it, the answer is respectively $x\mapsto 9$ or $\{x\mapsto x^2p(x)+9|p(x)\in\Bbb R[x]\}$.

J.G.
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