Abstract Algebra: Theory and Applications · Problem 21.8.2
Finite Fields — Thomas Judson
Mathematics — Abstract Algebra · splitting-field-quotient-ring-sage
Problem
This problem continues the previous one. Build the ring of polynomials over and within this ring use to generate a principal ideal. Finally construct the quotient of the polynomial ring by the ideal. Since the polynomial is irreducible, this quotient ring is a field, and by Proposition 21.1.12 this quotient ring is isomorphic to the number field in the previous problem. Borrowing from your results in the previous question, construct five roots of the polynomial within this quotient ring, but now as expressions in the generator of the quotient ring (which is technically a coset). Use Sage to verify that they are indeed roots. This demonstrates using a quotient ring to create a splitting field for an irreducible polynomial over a finite field.Intuitive explanation
The key insight is that when you quotient a polynomial ring by an irreducible polynomial's ideal, you're essentially adjoining a "formal root" of that polynomial—the coset of becomes a new element satisfying . Since is irreducible over , this quotient ring is actually a field where you can now express all roots as polynomials in this generator. Your job is to find those five roots by using the algebraic relations in the quotient ring, then verify they work by substituting back into and confirming you get zero.
Intuitive explanation
The key insight is that when you quotient a polynomial ring by an irreducible polynomial's ideal, you're essentially adjoining a "formal root" of that polynomial—the coset of becomes a new element satisfying . Since is irreducible over , this quotient ring is actually a field where you can now express all roots as polynomials in this generator. Your job is to find those five roots by using the algebraic relations in the quotient ring, then verify they work by substituting back into and confirming you get zero.