What are the invertible elements of the $K[X]$ ring where $K$ is a field?
I know that in a field all the elements are invertible but I'm having trouble linking $K$ and $K[X]$
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Hint: If a polynomial is invertible of degree $n$, what can the degree of the inverse be? – Tobias Kildetoft Jun 20 '16 at 10:23
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If we have two nonzero polynomials $f$ and $g$ of degrees $n$ and $m$ then consider the leading terms, say $aX^n$ and $bX^m$. The product $fg$ must have leading term $abX^{n+m}$. Since $K$ is a field we can't have $ab=0$ and so $fg$ has degree $n+m$. Thus $fg$ can't be $1$ unless both $f$ and $g$ have degree $0$.
Conversely it's clear that every nonzero polynomial of degree $0$ is invertible.
Oscar Cunningham
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