----- > [!proposition] Proposition. ([[characterization of quotienting a polynomial ring by a principal ideal]]) > Let $R$ be a nonzero [[ring]], and let $f(x)=x^{d}+a_{d-1}x^{d-1}+ \dots + a_{1}x + a_{0} \in R[x]$ > be a [[polynomial 4|polynomial]] that is [[monic polynomial|monic]] and of degree $d$. [^1] Then [[the division algorithm for polynomials]] lets us divide $f(x)$ with remainder as $g(x)=f(x)q(x)+r(x)$ > where $\text{deg }r(x) < \text{deg }f(x)$ and $q(x),r(x)$ are unique. > > Assume further that $R$ is [[commutative ring|commutative]]. In this case, the above is saying that for every $g(x) \in R[x]$ there exists a unique [[polynomial 4|polynomial]] $r(x)$ of $\text{degree}<\text{deg }f(x)$ such that $g(x)+\langle f(x) \rangle =r(x)+\langle f(x) \rangle $ > as [[coset|cosets]] of the [[principal ideal]] $\langle f(x) \rangle$ in $R[x]$. Thus the function $\varphi:R[x] \to R^{\oplus d}$ defined by sending $g(x)$ to the remainder of the division of $g(x)$ by $f(x)$ is [[well-defined]]. It induces an [[group isomorphism|isomorphism]] of [[abelian group|abelian groups]] $\frac{R[x]}{\langle f(x) \rangle } \cong^{\overline{\varphi}} R^{ \oplus d}.$ > Here, $\langle f(x) \rangle$ denotes the [[principal ideal]] of $f(x)$. > > Continuation: [[ring structure on direct sum induced by quotienting a polynomial ring by a principal ideal|Different isomorphisms allows us to endow]] $R^{\oplus d}$ with different [[ring]] structures. For example, the [[field]] structure of $\mathbb{C}$ can be realized in this manner if we set $R=\mathbb{R}$ and $f(x)=x^{2}+1$. ^proposition > [!proof]- Proof. ([[characterization of quotienting a polynomial ring by a principal ideal]]) > To begin, we claim $\varphi$ is a [[group homomorphism|homomorphism]] of [[abelian group|abelian groups]]. Indeed, if $g_{1}(x)=f(x)q_{1}(x)+r_{1}(x)$ and $g_{2}(x)=f(x)q_{2}(x)+r_{2}(x)$ with $\text{deg } r_{1}(x), \text{deg }r_{2}(x)<d$ then $g_{1}(x)+g_{2}(x)=f(x)(q_{1}(x) + q_{2}(x))+ (r_{1}(x)+r_{2}(x))$ with $\text{deg }(r_{1}(x) + r_{2}(x)) <d$ (by a [[polynomial 4#^properties|property here]]). Hence $\varphi\big( g_{1}(x)+g_{2}(x) \big)=r_{1}(x)+r_{2}(x)=\varphi(g_{1}(x))+\varphi(g_{2}(x)).$ Also, $\varphi$ is evidently a [[surjection]], for recall that any element of $R^{\oplus d}$ may identified with a polynomial of degree less than $d$ (such as a remainder following division by $f(x)$). Thus the [[first isomorphism theorem]] (for groups) yields an [[isomorphism]] $\frac{R[x]}{\text{ker }\varphi} \cong R ^{\oplus d}.$ We claim $\langle f(x) \rangle=\ker \varphi$. Indeed, $\varphi(g(x))=0$ if and only if $f(x)$ divides $g(x)$, i.e., if and only if $g(x)=f(x)h(x)$ for some $h(x)$ in $R[x]$, i.e., if and only if $g(x)$ is in the [[principal ideal]] [[ideal generated by a subset|generated by]] $f(x)$. ----- #### [^1]: This is not a substantial requirement if the [[polynomial 4|coefficient ring]] $R$ is a [[field]] (for then we may just 'divide out by the leading coefficient'), but could be if not: for example, $\langle 2x \rangle \subset \mathbb{Z}[x]$ cannot be [[ideal generated by a subset|generated by]] a [[monic polynomial]]. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```