-1

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

1 Answers1

0

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.

Wuestenfux
  • 20,232
  • 2
  • 12
  • 24