Irreducible polynomial means no roots?

An element $a$ of any ring (including polynomial rings) is reducible if and only if there exist elements $b$ and $c$ such that

• $a = bc$
• $b$ is not invertible
• $c$ is not invertible

In the special case of polynomial rings over fields, an element (i.e. a polynomial) $f$ is reducible if and only if there exist non-constant polynomials $g$ and $h$ such that $f = gh$. This is because the non-zero constants are precisely the invertible elements.

The condition of being irreducible if it doesn't have any roots is false. Consider, for example, the polynomial

$$ x^4 + 4 x^2 + 3 = (x^2 + 1)(x^2 + 3) \in \mathbb{R}[x] $$

When the coefficient ring is not a field, though, some coefficients are not invertible. The polynomial

$$ 2x \in \mathbb{Z}[x]$$

is reducible, because it is the product of $2$ and of $x$, both of which are not invertible. However, $2x \in \mathbb{Q}[x]$ is irreducible; the key difference is in this latter case, $2$ is invertible. Also, note that $2x$ has a rational root, despite being irreducible in $\mathbb{Q}[x]$.

What you're asking is almost true. An irreducible polynomial has a root if and only if it is linear. Proof:

Let $k$ be an integral domain. Assume that $f\in k[x]$ is irreducible, i.e. whenever $f=gh$, then either $g$ or $h$ is a unit. Assume that $a\in k$ is a root of $f$, i.e. $f(a)=0$. We perform polynomial division of $f$ by $(x-a)$, yielding $f=(x-a)g + r$ with $\deg(r)<\deg(x-a)=1$, meaning $r\in k$. Since $0=f(a)=r(a)$, it follows that $r=0$ and hence, $f=(x-a)g$ with $g\in k[x]$. But since $f$ is irreducible, this means that $g$ is a unit, i.e. $f$ is a linear polynomial.

I know there's an accepted answer here, but I just wanted to add in something to clarify a couple answers for newcomers:

If $F$ is a field, $f(x)\in F[x]$ is reducible if and only if $f(x)$ has a zero in $F$, but this is only always true for polynomials of degree 2 and 3.

Mark Bennet gives a decent counterexample to the generalized claim, and note that the polynomial he uses is degree 4.

However, things are a bit different when you're working in $Z_n$ ($Z/nZ$). You can check for reducibility by testing if $f(n)=0$ for $n\in[0,n-1]$.

For example, $f(x)=x^3+1\in Z_9[x]$ is reducible over $Z_9$ because $f(2)=0$.