A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same integral domain. > [!NOTE] Definition 1 (Irreducible Polynomial) > Let $D$ be an [[Integral Domain|integral domain]]. Let $D[x]$ be the [[Ring of Polynomial Forms|ring of polynomials]] over $D$ in $x$ ([[Ring of Polynomial Forms over Integral Domain is Integral Domain|which is an integral domain]]). Let $f\in D[x].$ Then $f$ is an irreducible polynomial over $D$ in $x$ if it is a non-constant polynomial (neither a unit nor zero, $\deg(f)>0$) that is an [[Irreducible Elements of Integral Domain|irreducible element]] of $D[x],$ that is: for all $g,h\in D[x],$ if $gh=f$ then $g$ or $h$ is a [[Unit in a Ring|unit]] of $D[x].$ > [!NOTE] Definition 2 (Irreducible Polynomial) > An irreducible polynomial over $D$ in $x$ is a non-constant polynomial that is not the product of two polynomials of smaller degree. **Note** that the above definitions are equivalent: if $f=gh$ such that degrees of both $g$ and $h$ are less than that of $f.$ Then neither the degree of $g$ nor $h$ can be zero since $\deg(g)+\deg(h)=\deg(f)\geq1$ and if WLOG $\deg(g)=0$ then $\deg(h)=\deg(f)$ contradicting the fact that the degree of $h$ is less than the degree of $g.$ The converse is similarly true. > [!Example] Examples > Any degree one polynomial in $x$ over a Field $F$ is irreducible over $F$: suppose $\deg(f)=1$ and $f=gh.$ Then by [[Degree of Product of Polynomials Over Integral Domain]], $1=\deg(g)+\deg(h)$ which implies that either the degree of $g$ or $h$ is zero and by [[Units of Ring of Polynomial Forms over Integral Domain|units in ring of polynomials over a field]], it is a unit in $F[x].$ # Properties **Irreducibility in $\mathbb{C}[x]$ and $\mathbb{R}[x]$**: 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 $b^{2}-4ac<0.$ **Irreducibility in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$**: First 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 $\mathbb{Q}.$ # Applications **UFDs**: Note that any non-constant univariate polynomial over a field can be uniquely expressed as a product of irreducible polynomials over the same field up to multiplication by units ([[Ring of Polynomial Forms over Field is Unique Factorisation Domain|F[x] is a unique factorisation domain]]).