> [!NOTE] Theorem (Division with Remainder Theorem for Univariate Polynomial Forms over Field) > Let $F$ be a [[Field (Algebra)|field]]. Let $F[x]$ be the [[Ring of Polynomial Forms|ring of polynomial forms]] over $F$ in $x.$ For any $f\in F[x],$ let $\deg(f)$ denote the [[Degree of a Polynomial|degree]] of $f.$ Then for all $f,g\in F[x] \setminus \{ 0 \},$ there exists unique $q,r\in F[x]$ such that $f=gq +r$with $\deg(r)<\deg(g).$ **Note** that the statement of this theorem is closely related to the statement of [[Division with remainder for integers]]. Key idea [[Ring of Polynomial Forms over Field is a Euclidean Domain]]. *Proof of existence by induction*: Fix $g\in F[x]\setminus\{ 0 \}.$ For $k\in \mathbb{N},$ let $P(k)$ be the proposition: for all $f\in F[x]$ with $\deg(f)<k,$ there exists $q,r\in F[x]$ such that $f=gq+r$ with $\deg(r)<\deg(g).$ ~~(**Skip**: If $\deg(f)=0$ then $f=a_{0}$ for some $a_{0}\in F^{*}.$ If $\deg(g)\geq 1$ then $f=0\times g+a_{0}$ and $0=\deg(f)<\deg(g).$ If $\deg(g)=0$ then $g=a_{1}$ for some $a_{1}\in F^{*}$ and $f=a_{0}a_{1}^{-1}a_{1}+0$ with $-\infty=\deg(0)<\deg(a_{1})=0.$ So $P(1)$ is true.)~~ If $f=0,$ then take $q=r=0$ thus $P(0)$ is true since $g \neq 0 \implies\deg(g) \geq 0$ and $\deg(0)=-\infty.$ Suppose for some $n\in \mathbb{N},$ $P(n)$ is true. Let $f\in F[x]$ such that $\deg(f)=n.$ Let the leading coefficients of $f$ and $g$ be $a_{n},b_{m}\in F^{*}$ respectively. If $\deg(f)<\deg(g)$ then take $q=0,r=f.$ Otherwise we can subtract from $f$ a suitable multiple of $g$ so as to eliminate the highest term in $f$: if $\deg(f)\geq\deg(g),$ consider $h:=f-a_{n}b_{m}^{-1}x^{n-m}g$Then $\deg(h)<n$ since the coefficient of $x^{n}$ in $h$ is $0.$ Therefore by $P(n),$ $h=gq_{0}+r$ where $\deg(r)<\deg(g).$ Therefore, $f=g(q_{0}+a_{n}b_{m}^{-1}x^{n-m}g) + r$and $P(n+1)$ is true. Thus by [[Induction Principle|mathematical induction]], $P(n)$ is true for all $n\in \mathbb{N}.$ *Proof of uniqueness*: Suppose $gq+r=gq'+r'$ with $\deg(r),\deg(r')<\deg(g).$ Then $g(q-q')=r'-r$By [[Degree of Sum of Polynomials]], $\deg(r'-r)\leq \max\{ \deg(r'), \deg(r) \}<\deg(g)$By [[Degree of Product of Polynomials Over Integral Domain]], $\deg(g(q-q'))=\deg(g)+\deg(q-q')$Thus $\deg(g)<\deg(g) + \deg(q-q')$which implies $\deg(q-q')=-\infty$ that is $q=q'.$ Also $r'-r=g(q-q')=0$ and so $r=r'.$ # Applications **Consequences**: The [[Polynomial Factor Theorem|factor theorem]] asserts that $f(\alpha)=0_{}$ for some $\alpha\in F$ if and only if $f=(x-\alpha)q$ for some $q\in F[x].$ The [[Ring of Polynomial Forms over Field is a Principal Ideal Domain|ring of polynomial forms over a field is a principal ideal domain]] meaning that every ideal has the form $fF[x]=\{ fg \mid g\in F[x] \}$ for some $f\in F[x].$ The [[Polynomial Remainder Theorem|remainder theorem]] asserts that $f(\alpha)$ is the remainder of $f$ after division by $(x-\alpha).$