Abstract Algebra Vs. Complex Analysis
Here's the dirt on how these objects relate to one another:
The very definition of the complex numbers traditionally comes from abstract algebra: you take the ring of real polynomials, mod out by the ideal generated by x^2+1, and since x^2+1 is irreducible, theorems of algebra guarantee the resulting quotient ring is actually a field-- a field with a square root of -1. Complex analysis generously repays the favor by providing an unexpected proof of the Fundamental Theorem of Algebra (which states that every non-constant polynomial with complex coefficients has a root).