Post #2141603

2025-04-06 21:22 UTC

This criterion of Eisenstein requires an auxiliary prime number, and I'll state it as an example, with another polynomial, say T ^ 4 - 10 T ^ 2 + 2. Here, one takes the prime number 2, and one observes that modulo 2, the polynomial is equal to T ^ 4, while the constant term, 2, is not divisible by 2^2=4. In that case, the criterion immediately asserts that the polynomial T^4-10T^2+2 is irreducible.

Replies (1)

@antoinechambertloir@mathstodon.xyz 2025-04-06 21:26

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.

Open ##2141604