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My lecture notes state that an 'easy' result is

If $R$ is an integral domain then an irreducible element of $R$ remains irreducible in $R[x]$, and the units in $R$ and in $R[x]$ are the same.

I can't seem to get my head around why this is the case, and what a unit in $R[x]$ means intuitively because I don't see how the units can be the same if $R$ is the coefficients of the polynomials in $R[x]$. I.e. for an unit say $\alpha \in R$ then what is the 'corresponding' unit in $R[x]$? I is it $\alpha x$ or $\alpha x^2$... or am I getting the wrong end of the stick here?

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    $R$ embeds in $R[x]$ as the constant polynomials. A unit $\alpha$ in $R$ corresponds to the constant polynomial $\alpha$, which is a unit in $R[x]$.2012-05-14
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    [See here](http://math.stackexchange.com/a/19145/242) on units in general polynomial rings.2014-04-17
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    If $f(x)$ is a unit and if $g(x)$ is its inverse, then $${\rm deg}(f(x)g(x)) = {\rm deg}(1) = 0.$$ This forces ${\rm deg}(f) ={\rm deg}(g) = 0.$ In other words a polynomial ring in one variable whose coefficients are drawn from an integral domain can have no units that are not constants.2018-01-24

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