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Invertible Polynomial


A polynomial admitting a multiplicative inverse. In the polynomial ring R[x], where R is an integral domain, the invertible polynomials are precisely the constant polynomials a such that a is an invertible element of R. In particular, if R is a field, the invertible polynomials are all constant polynomials except the zero polynomial.

If R is not an integral domain, there may be in R[x] invertible polynomials that are not constant. In Z_4[x], for instance, we have:

 (2^_x+1^_)(2^_x+1^_)=1^_,

which shows that the polynomial 2^_x+1^_ is invertible, and inverse to itself.


See also

Invertible, Invertible Element, Invertible Polynomial Map

This entry contributed by Margherita Barile

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Barile, Margherita. "Invertible Polynomial." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InvertiblePolynomial.html

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