SUMMARY
The discussion focuses on the factorization of the polynomial x³ - 23x² - 97x + 291 over finite fields Z3[x], Z7[x], and Z11[x]. The polynomial is shown to be reducible over Z3, where it factors into X(X² + X + 2), but the quadratic X² + X + 2 has no roots in Z/3Z. In Z7, the polynomial is confirmed to be irreducible as it has no roots. Similarly, in Z11, the polynomial is also irreducible after testing potential roots, confirming that it does not factor over this field.
PREREQUISITES
- Understanding of polynomial factorization
- Familiarity with finite fields, specifically Z3, Z7, and Z11
- Knowledge of root testing in modular arithmetic
- Basic algebraic manipulation skills
NEXT STEPS
- Study polynomial factorization techniques over finite fields
- Learn about irreducibility criteria for polynomials
- Explore root-finding algorithms in modular arithmetic
- Investigate applications of finite fields in coding theory
USEFUL FOR
Mathematicians, students studying abstract algebra, and anyone interested in polynomial factorization over finite fields.