The univariate polynomial over a field of characteristic zero has a nontrivial factor with its first derivatives. The converse of this result has been conjectured by Casas-Alvero.
The authors show that the conjecture is true for some cases of . More precisely, they show that, over a field of characteristic zero, for or , a prime number, the only degree univariate monic polynomial that has nontrivial factors with its first first derivatives is .
For fields of positive characteristic , the authors present the counter-example of degree which is not a th power and has nontrivial common factors with its first Hasse derivatives, which is a stronger condition than regular derivatives.