SUMMARY
The polynomial (x-a1)(x-a2)...(x-an)+1 is proven to be irreducible in Z[x] when a1, a2, ..., an are distinct odd integers. The discussion highlights the failure of the Eisenstein criterion in this context, as demonstrated by the counterexample (x-1)(x-3)+1, which simplifies to (x-2)^2, indicating reducibility. This counterexample serves to clarify the conditions under which the polynomial remains irreducible.
PREREQUISITES
- Understanding of polynomial irreducibility in Z[x]
- Familiarity with the Eisenstein criterion for irreducibility
- Knowledge of polynomial factorization techniques
- Basic concepts of distinct integers and their properties
NEXT STEPS
- Study the Eisenstein criterion in depth to understand its limitations
- Explore additional counterexamples to polynomial irreducibility in Z[x]
- Learn about other irreducibility tests applicable to polynomials
- Investigate the properties of polynomials with integer coefficients
USEFUL FOR
Mathematics students, algebra enthusiasts, and anyone studying polynomial theory and irreducibility in number theory.