Advanced Algebra II continues the structural approach to mathematics by focusing on three fundamental algebraic systems: the integers, complex numbers, and polynomials. [cite_start]The course begins with a deep dive into the ring of integers $\mathbb{Z}$, covering the division algorithm, the Euclidean algorithm, prime numbers, unique factorization, and the theory of congruences (modular arithmetic)[cite: 338].
We then transition to the construction and properties of the field of complex numbers $\mathbb{C}$. [cite_start]Students explore their geometric representation, polar form, and key results such as De Moivre’s theorem and the extraction of $n$-th roots[cite: 338].
The final section of the course generalizes arithmetic concepts to the ring of polynomials over a field. [cite_start]We study divisibility, reducibility, and factorization of polynomials, culminating in the proof and applications of the Fundamental Theorem of Algebra, which asserts that every non-constant polynomial has a root in $\mathbb{C}$[cite: 338, 347].
This document compiles selected notes and examples developed during the course. While not intended as a comprehensive textbook, these materials provide detailed explanations of key topics and are designed to serve as a robust supplementary resource for students.
As with the notes, this section contains a compilation of various homework assignments and problem sets. Although they do not follow a unified structure, they may serve as valuable practice material for anyone who wishes to reinforce their understanding of the topics covered in the course.
