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Are there any general steps or things to look out for when identifying the unit elements, irreducible elements and prime elements in ring of polynomials e.g. $\mathbb{Z}_4[x]$, $\mathbb{C}[x]$, $\mathbb{R}[x]$, $\mathbb{Q}[x]$.

It seems a little tricky.

Thank you.

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  • $\begingroup$ It gets tricky, yes! $\endgroup$ – Andrea Mori Apr 6 at 10:34
  • $\begingroup$ Be aware that there in non-domains the notions of "associate" and "irreducible" bifurcate into a few inequivalent notions, e.g. see here, so it is essential to state which definition one is using. $\endgroup$ – Bill Dubuque Apr 6 at 13:49
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The first one is tricky. The other three are well-known First of all units can only be from polynomials that are constants.

For irreducible polynomials over $\mathbf Q$ one has well-known Eisenstein theorem that guarantees one in each degree (alternately simpler result about the integers: like $\sqrt[n]p$ is irrational for any prime number shows that the polynomial $X^n-p$ is irreducible for every $n$.)

For complex numbers we use Liouville's theorem and conclude that all polynomials have roots, allowing us to conclude that irreducible ones have to be of degree 1. From this one can conclude that irreducible real polynomials have to be either in degree 1 or 2, and not higher. Primes and irreducibles are the same in them is a consequence of the fact that this is a PID.

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