One of the irreducible criteria is well-known for polynomials with integers coefficients and also holds naturally for polynomials of is Eisenstein's irreducible criterion that states
Criterion 1 (Eisenstein's irreducible criterion).
Let . If there exists a prime number satisfying the following three conditions:
- divides each for . - does not divide , and - does not divide .
Then the polynomial is irreducible.
We have another criterion, which is even easier to check.
Criterion 2: Let . Suppose that and is a prime number, then is irreducible.
Proof
Indeed, suppose that for some . Note that , and since is prime, we know that either or . WLOG, assume that , then for some . Note that if , then , and then , which contradicts the hypothesis. Therefore , and this implies that is irreducible.
A New CrowdMath Project (with Resource 0 and Some Exercises)
felixgotti2
NOct 15, 2024
by felixgotti
Hi everyone!
A New Research Project for CrowdMath 2024 has been released! This new research project will be about semidomains satisfying the statement of the Goldbach Conjecture.
We have the great pleasure to have Dr. Harold Polo with us, providing his direct help with this project, which is in turn motivated by the current research carried out by Dr. Polo in the intersection of semidomains and the statement of the Goldbach Conjecture.
Resource 0 (with some initial exercises) has just been posted. As always, I hope you enjoy learning the new material and working on the exercises. We plan to post the first open problem by next week.
Don't hesitate to bring your questions, solutions, and ideas to the message board!