Computing Z[x]/⟨x² + 1⟩

Abstract Algebra · Axiom Academy

EXAMPLE Computing Z[x]/⟨x² + 1⟩ Discover the isomorphism with Gaussian integers through polynomial reduction and coset representatives Excellent work! You've constructed the quotient ring Z[x]/⟨x² + 1⟩ and verified its isomorphism with Z[i]. Here's what we learned: The Ideal I = ⟨x² + 1⟩: Contains all polynomials of the form (x² + 1)·g(x). Working in Z[x]/I means we can replace x² with -1. Coset Representatives: Every element in Z[x]/⟨x² + 1⟩ can be uniquely written as a + bx where a, b ∈ Z. This is the standard form. x Behaves Like i: Since x² ≡ -1 (mod I), the element x in the quotient ring acts exactly like the imaginary unit i. Isomorphism φ: Z[x]/⟨x² + 1⟩ → Z[i]: Given by φ(a + bx + I) = a + bi. This is a ring isomorphism preserving addition and multiplication. Why This Works: The relation x² = -1 in the quotient ring perfectly captures the defining property of i, making the two structures algebraically identical. This construction demonstrates how quotient rings can create new algebraic structures from polynomial rings. The same technique works for other quotients: Z[x]/⟨x² + x + 1⟩ gives Z[ω] where ω is a primitive cube root of unity!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.