# Definitions > [!NOTE] Definition (Ring of Univariate Polynomial Forms) > Let $R$ be a [[Commutative Ring|commutative ring with unity]]. The *ring of polynomial forms* over $R$ in $x$ is given by the [[Sets|set]] $R[x]=\{ a_{n} x^{n} + a_{n-1}x^{n-1} + \dots+a_{1}x+a_{0} \mid a_{i} \in R, n\in \mathbb{N} \}$ > [!Example] > The [[Zero Polynomial|zero polynomial]] is the same as the zero of $R.$ Its degree is defined as $-\infty.$ # Properties **Degree**: The [[Degree of a Polynomial|degree]] of a non-zero polynomial in one indeterminate is the largest $k\in \mathbb{N}$ such that the coefficient of $x^{k}$ is non-zero. The [[Degree of Product of Polynomials|degree of the product of non-zero polynomials is less than or equal to the sum of degrees]]. **General algebraic properties**: We define [[Polynomial Addition|polynomial addition]] and [[Polynomial Multiplication|polynomial multiplication]] such that $(R[x_{1},\dots,x_{n}],\times,+)$ is [[Ring of Polynomial Forms is a Ring|indeed a ring]]. **Univariate polynomials over integral domains**: Note that [[Degree of Product of Polynomials Over Integral Domain|degree of the product of polynomials over an integral domain equals the sum of their degrees]]: let $D$ be an integral domain and $f,g\in D[x]$ then $\deg(fg)=\deg(f)+\deg(g).$ It follows that if $D$ is an integral domain then so is $D[x]$ ([[Ring of Polynomial Forms over Integral Domain is Integral Domain|D[x] has no proper zero divisors]]). **Univariate polynomials over fields**: The [[Units of Ring of Polynomial Forms over Integral Domain|units of the ring of polynomial forms over a field]] are those with degree zero (non-zero constant polynomials). The [[Division with Remainder Theorem for Ring of Polynomial Forms over Fields|division with remainder theorem for polynomial forms over field]] asserts that for any $f$ and $g$ that are non-zero polynomials in $x$ over a field, there exists $q$ and $r$ such that $f=gq+r$ and $\deg(r)<\deg(g).$ More formally, [[Ring of Polynomial Forms over Field is a Euclidean Domain|a ring of polynomial forms over a field is a Euclidean domain]] just like the [[Comparison of Euclidean Domains - The Integers vs Polynomial Ring over Field|integers]]. A corollary of this is the [[Polynomial Factor Theorem|factor theorem]] which asserts that $f(\alpha)=0_{}$ for some $\alpha\in F$ if and only if $f=(x-\alpha)q$ for some $q\in F[x]$ where $F$ is a field. Another is that [[Ring of Polynomial Forms over Field is a Principal Ideal Domain|ring of polynomial forms over a field is a principal ideal domain]] that is: every ideal of $F[x]$ is generated by the a single element of the ideal through multiplication. Finally, [[Ring of Polynomial Forms over Field is Unique Factorisation Domain|a ring of polynomial forms over a field is a unique factorization domain]] and in general every Euclidean domain is a UFD because we have [[Euclid's Lemma for Irreducible Polynomial Forms over Field|Euclid's lemma]]. **Reducibility tests for univariate polynomials over number fields**: An [[Irreducible Polynomial|irreducible polynomial]] over an integral domain $D$ is a non-constant polynomial that is not the product of two non-constant polynomials in $D[x].$ By the [[Fundamental Theorem of Algebra|fundamental theorem of algebra]], the [[Irreducible Polynomials Over The Complex Numbers|irreducible polynomials over the complex numbers]] are exactly those of degree $1$ and the [[Irreducible Polynomials Over The Real Numbers|irreducible polynomials over the real numbers]] are exactly those of degree $1$ or those of degree $2$ of the form $ax^{2}+bx+c$ for some $a,b,c\in \mathbb{R}$ where $b^{2}-4ac<0.$ Note that [[Polynomial which is Irreducible over Integers is Irreducible over Rationals|irreducibility over Z implies irreducibility over Q]]. However, the converse is not true: for example $3x+3$ is reducible over $\mathbb{Z}$ since neither $3$ nor $x+1$ are units in $\mathbb{Z}[x]$ but is irreducible over $\mathbb{Q}$ by the argument in the example and the fact that $\mathbb{Q}$ is a field. [[Schönemann-Eisenstein Criterion|Eisenstein's criterion]] gives a sufficient condition for a polynomial in $x$ over $\mathbb{Z}$ to be irreducible over $\mathbb{Z}$ (and thus $\mathbb{Q}$). **Relations to quotient rings**: ...