“Advanced Algebra” is a comprehensive textbook that covers advanced topics in algebra for students and professionals.
1- Abstract Algebra: This section introduces abstract algebra, including groups, rings, and fields. It covers the axiomatic definitions and theorems, and provides examples of each concept.
2- Polynomials: This section covers polynomials, including the degree, leading coefficient, roots, and factorization of polynomials. It also covers the Fundamental Theorem of Algebra, and provides methods for solving polynomial equations.
3- Galois Theory: This section introduces Galois theory, which studies the connection between algebraic equations and their solutions. It covers the Galois group, normal subgroups, and solvability criteria, and provides examples of applications of Galois theory.
4- Modular Arithmetic: This section covers modular arithmetic, including the modular congruences and the Chinese Remainder Theorem. It also covers the Euler Totient Function, the Fermat Little Theorem, and the RSA encryption algorithm.
5- Linear Algebra: This section covers linear algebra, including vectors, matrices, determinants, and eigenvalues. It also covers linear transformations, orthogonality, and the singular value decomposition.
6- Algebraic Geometry: This section introduces algebraic geometry, which studies the geometric properties of algebraic equations. It covers curves, surfaces, and varieties, and provides methods for finding the solutions to algebraic equations geometrically.
7- Commutative Algebra: This section covers commutative algebra, including ideals, prime ideals, and maximal ideals. It also covers the Nullstellensatz, primary decomposition, and integral extensions.
8- Conclusion: This section summarizes the key points covered in the book, and provides recommendations for further learning and practice.
“Advanced Algebra” is a comprehensive textbook that provides a deep understanding of the theory and applications of algebra. The clear explanations, concise examples, and hands-on exercises will help you build a strong foundation in advanced algebra, and the accompanying code samples will help you apply what you have learned to real-world problems. Whether you are a student, a professional, or simply interested in advancing your understanding of algebra, this book is an essential resource.