Post #2141604

2025-04-06 21:26 UTC

With all due respect to Eisenstein, I don't like that criterion too much, though, because it only applies in kind of exceptional situations. Still, it is often useful. There is a classic case of application, by the way, of the Eisenstein criterion, namely the irreducibility of *cyclotomic* polynomials of prime index. For example, T^4+T^3+T^2+T+1 (for p=5). But it is not visible that the criterion applies, and one needs to perform the change of variable T = X+1. I had noticed that in that case, it maybe interesting to generalize the Eisenstein criterion to avoid this change of variables.

Replies (1)

@antoinechambertloir@mathstodon.xyz 2025-04-06 21:31

Indeed, for a monic polynomial f in Z[T], a variant of the criterion states: take an integer a and a prime number p, and assume that : * f(T) = (T-a)^d mod p * f'(a) (derivative) is not divisible by p^2. Then f is irreducible. For the above polynomial, f(T) = T^4+T^3+T^2+T+1, still taking p=5, one has f(T) = (T-1)^4 mod 5 and f'(1)=4·5/2=10. The reason, that avoids any strange computation, is that f(T)=(T^5-1)/(T-1), which modulo 5 is (T-1)^4 (by the divisibility of the binomial coefficients).

Open ##2141605