Abstract Algebra 29A: Visualize Factor Rings of Gaussian Integers Z[i], Construct Complex Field C

If A is the principal ideal generated by x^2+1, then R[x]/A is isomorphic to the field of complex numbers C. Here, R[x] is the ring of polynomials with coefficients in the real number field R. Therefore, we can construct the field C as a factor ring.

Abstract Algebra, Lecture 29A.

(0:00) Announcements.
(3:46) The Gaussian integers Z[i] form a Euclidean Domain.
(12:53) Mathematica visualization of factor rings in the Gaussian integers. Example 1: Z[i]/A, where A is the principal ideal generated by 1 + i.
(20:37) Example 2: Z[i]/A, where A is the principal ideal generated by 2 + i.
(27:35) Example 3: Z[i]/A, where A is the principal ideal generated by 2 + 2i.
(29:17) Review facts about polynomial rings and factor rings.
(32:38) Idea of how to show R[x]/A, where A is the principal ideal generated by x^2 + 1, is isomorphic to the complex numbers C (in fact, it can be thought of as one way to "construct" C).

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