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In the other direction, if (f(x)) is reducible and has degree 2, then it is easy to see that it must have a monic, linear right factor.
Conversely, if (x-t) is a (right) factor of (f(x)), then (f(x) = h(x)(x-t)) for some (h(x) inmathsf{K}[x; sigma]).
Then (x-t) is a right factor of (f(x)) if and only if t is a σ-root of (f(x)).
When (f(x) = q(x)(x-t)), instead of simply saying that (x-t) is a factor of (f(x)), we may sometimes use the more precise terminology that (x-t) is a (right) factor of (f(x)).
Therefore, if t is a σ-root of (f(x)), then (f(t; sigma) = 0), and we can see that (x-t) is certainly a (right) factor of (f(x)).
One direction follows immediately from Theorem 2.3, for if (f(x)) has a σ-root in K, then it has a right factor of degree one in (mathsf{K}[x; sigma]) and is therefore reducible.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com