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When does a polynomial in the ring of polynomial have an inverse? I thought only constant polynomials were units. if there are other units, under what rings can we guarantee the existence of inverse for non-constant polynomials

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  • $\begingroup$ What about non-commutative rings $\endgroup$ – vinolyn sylvia Feb 8 at 7:28
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Well, its an argument involving degrees.

If $f,g$ are polynomials with $f\cdot g=1$, then $$0 = {\rm deg}(1) = {\rm deg}(f\cdot g) = {\rm deg}(f) + {\rm deg}(g).$$ The last equality holds if the leading coefficient don't cancel. This holds in integral domains. Then the degrees of $f$ and $g$ must be zero. Thus $f,g$ are constant polynomials.

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  • $\begingroup$ What about rings which are not integral domains $\endgroup$ – vinolyn sylvia Feb 8 at 7:36
  • $\begingroup$ For polynomial rings over any non integral domain the units in the polynomial ring are precisely the units in the integral domain. $\endgroup$ – Dbchatto67 Feb 8 at 7:58

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