- Published Year: 1993
- Page count: 425
- File Size: 2 Mb
- Language: English
- Published by: Springer
- Visited by: 0
- Rating/Review: 4.5
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ISBN:
0387940901
9780387940908
Keywords:
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Each download or ask from book AI costs 2 points. To earn more free points, please visit the Points Guide Page and complete some valuable actions.Introduction to 'Algorithmic Algebra'
Welcome to the intricate world of 'Algorithmic Algebra', a profound exploration into the realms of algorithms intersecting with algebra. Intended for both academics and professionals, this book offers a detailed insight into how algebraic concepts can be perceived and manipulated using algorithmic techniques. The intersection of these fields provides innovative perspectives and solutions to classical problems, redefining the approach to both education and application in these areas.
Detailed Summary of the Book
In 'Algorithmic Algebra', we commence our journey with foundational concepts that lay the groundwork for more complex ideas. The book starts with an introduction to basic algebraic structures, such as groups, rings, and fields, alongside a review of essential algorithms that perform fundamental operations on these structures. As the chapters progress, readers will encounter polynomial ideals and Gröbner bases, which serve as pivotal concepts for solving systems of polynomial equations.
The narrative enhances through the discussion on algorithmic methods for solving polynomial equations, examining both classical and contemporary techniques. The book further delves into canonical forms, exploring algorithmic methods that simplify the representation of algebraic objects. This includes algorithms for computing characteristic polynomials and exploring eigenvalue-related problems.
Later chapters present discussions on algebraic coding theory and cryptography, illuminating how algebraic algorithms contribute to real-world applications in secure communications and data protection. Advanced topics such as computational number theory and algebraic geometry are also addressed, offering readers insights into more specialized applications and current research frontiers.
Key Takeaways
- Understanding the synergy between algorithms and algebra, enabling innovative problem-solving approaches.
- Knowledge of polynomial ideals and Gröbner bases, essential tools in computational algebra.
- Insight into algorithmic methods that aid in simplifying and computing forms of algebraic expressions.
- Comprehension of applications in cryptography and coding theory, highlighting the relevance of algebraic algorithms.
- A pathway to advanced studies in computational number theory and algebraic geometry.
Famous Quotes from the Book
"The beauty of algebra lies not just in its abstract structures, but in how these can be elegantly manipulated and solved through the lens of an algorithm."
"Algorithms serve as the bridge between theoretical concepts and practical applications, transforming algebra from a descriptive science into a constructive tool."
Why This Book Matters
'Algorithmic Algebra' is not merely a textbook; it is a comprehensive exploration that bridges theoretical foundations with computational techniques. This publication matters because it uniquely captures the convergence of two pivotal mathematical domains, showing how each enhances the understanding and application of the other. It is essential reading for anyone interested in the increasingly algorithm-driven world of modern mathematics. Understanding these concepts is vital for pursuits in academic research, advanced algorithm design, and applications in technology sectors where security and efficiency are paramount.
Moreover, the book aids in cultivating a mindset that values both abstraction and execution. In an ever-evolving digital landscape, mastery of these skills equips readers with the tools necessary for both innovative exploration and practical application, making significant contributions to both academia and industry.
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