Why is $x^3 + 3x + 2$ irreducible by plugging in elements of $\mathbb{Z}_5$

Question

The polynomial $x^3+3x+2$ is irreducible in $\mathbb{Z}_5[x]$. I get that it must take the form of $(x-a)g(x)$ where $a$ is a zero,but my book plugged in elements of $\mathbb{Z}_5$ to show no zeros, but I thought the indeterminate $x$ can take any value, not just $\mathbb{Z}_5$.

Answer 1

You are seeking a monic factor of your polynomial of degree $1$ in $\mathbb Z_5[x]$. This means a polynomial of the form $x-a$ for some $a\in \mathbb Z_5$.

It is certainly possible for $\mathbb Z_5$ to be included in a ring where this polynomial has a root. That doesn't contradict this polynomial being irreducible in $\mathbb Z_5[x]$.

For example:

• $x^2+1$ is irreducible in $\mathbb R[x]$, even though it has roots in $\mathbb C$.
• $x^2-2$ is irreducible in $\mathbb Q[x]$ even though it has a root in $\mathbb R$.
• $5$ is irreducible in the integers, but $5=(2+i)(2-i)$ is a factorization in the Gaussian integers.

Being irreducible in one ring $R$ does not imply being irreducible in all other rings that contain (an image of) $R$.