What is the significance of considering characteristic $p$ dividing $n(n-1)$, in finding the Galois group of a equation of degree $n$

Question

I read in this paper by K. Uchida, that, in finding Galois group of an equation $X^n-aX+b=0$, in theorem 1, author considers characteristic $p$ is not a divisor of $n(n-1)$.

I did not understand, this condition, p not dividing $n(n-1)$. Can any one explain this, Thank you very much.

Answer 1

Since the characteristic $p$ is prime, it means we have both $p\not\mid n$ and $p\not\mid (n-1)$, by Euclid's lemma.

Note, the author explains that other than in this situation his results have been limited... (to the cases $n=p^m$ and $n=p^m+1$).

Note also that a large class of rings (namely those with no nontrivial zero divisors) have characteristic $0$ or prime...

To find additional details, read the paper...