@antoinechambertloir@mathstodon.xyz
Post #2141602
2025-04-06 21:20 UTC
This is both a complicated and a trivial question.
Trivial because there are general bounds (initially due to Mignotte) for the integers that appear in any such factorization, and it could just be sufficient to try any possibility in the given range and conclude. Brute force, not very intelligent, but with a certain outcome.
Complicated because those computations would often be long, and there are many criteria in number theory to assert irreducibility.
One of the easiest to apply is the aforementioned Eisenstein criterion.
Replies (1)
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@antoinechambertloir@mathstodon.xyz 2025-04-06 21:22
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.