3 ms·
> If the polynomial P is invertible, i.e. has degree 1 Should be degree 0: only constant polynomials are invertible. E.g. x+1 is not invertible, and modding it
by red_trumpet 4y ago
> If the polynomial P is invertible, i.e. has degree 1
Should be degree 0: only constant polynomials are invertible. E.g. x+1 is not invertible, and modding it out doesn't result in the zero ring.
The example is a bit confusing, because $x=x+1$ is equivalent to $0=1$, which has degree 0.
- heinrichhartman 4y agoYes! Fixing
- hgsgm 4y agoWhat's wrong with negative degree monomials?
- red_trumpet 4y agoYou mean like $x^{-1} = 1/x$? That's called a rational function[1], but not a polynomial, so it's not an element of the polynomial ring[2]. Of course you can also consider the algebra of rational functions, but this is a field[3] (almost by definition: you make every polynomial invertible), which means that modding out anything other than 0 yields the zero ring[4]. [1] https://en.wikipedia.org/wiki/Rational_function https://en.wikipedia.org/wiki/Rational_function [2] https://en.wikipedia.org/wiki/Polynomial_ring https://en.wikipedia.org/wiki/Polynomial_ring [3] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) [4] https://en.wikipedia.org/wiki/Zero_ring https://en.wikipedia.org/wiki/Zero_ring