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A Course in Computational Algebraic Number Theory summary
Henri Cohen
Brief summary
A Course in Computational Algebraic Number Theory by Henri Cohen provides a comprehensive introduction to the computational aspects of algebraic number theory, covering topics such as factoring, primality testing, and algorithms for solving Diophantine equations.
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Table of Contents
A Course in Computational Algebraic Number Theory
Summary of key ideas
Exploring the Foundations
In A Course in Computational Algebraic Number Theory by Henri Cohen, we embark on a journey through the foundations of computational algebraic number theory. Cohen begins by introducing us to the basic concepts of number theory and algebraic number fields. He explains how these fields are essential in cryptography, coding theory, and other areas of computer science.
Next, Cohen delves into the world of algebraic integers and their properties. He explains how these integers are crucial in the study of algebraic number fields and their applications in cryptography. He then introduces us to the concept of ideal numbers and their significance in algebraic number theory.
Understanding Class Field Theory
After laying the groundwork, Cohen moves on to explore class field theory. He explains the deep connections between number theory and the theory of algebraic functions. He introduces us to the concept of global fields and their ring of adeles, providing the necessary background for the study of class field theory.
Class field theory is a central topic in computational algebraic number theory, and Cohen covers it in great detail. He discusses the reciprocity laws, the Artin map, and the concept of class groups, providing a comprehensive understanding of these fundamental concepts.
Exploring Computational Aspects
As we progress, Cohen transitions to the computational aspects of algebraic number theory. He introduces us to the algorithms used in this field, such as the Euclidean algorithm, the continued fraction algorithm, and the LLL algorithm. He explains how these algorithms are employed in solving problems related to algebraic number fields.
Cohen also discusses the use of elliptic curves in cryptography and their role in factorization algorithms. He explores the concept of modular forms and their applications, shedding light on their significance in computational algebraic number theory.
Applications in Cryptography
The latter part of A Course in Computational Algebraic Number Theory focuses on the applications of algebraic number theory in cryptography. Cohen provides a detailed overview of public-key cryptography, explaining how number-theoretic problems form the basis for secure cryptographic systems.
He discusses the RSA cryptosystem, the discrete logarithm problem, and the elliptic curve cryptosystems, illustrating how these systems rely on computational algebraic number theory. He also touches upon the role of algebraic number theory in ensuring the security of digital signatures and key exchange protocols.
Conclusion: A Comprehensive Journey
In conclusion, A Course in Computational Algebraic Number Theory takes us on a comprehensive journey through the theoretical underpinnings and practical applications of computational algebraic number theory. Cohen's meticulous exploration equips us with a deep understanding of the subject, making this book an invaluable resource for students, researchers, and practitioners in the field of number theory, cryptography, and computer science.
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What is A Course in Computational Algebraic Number Theory about?
A Course in Computational Algebraic Number Theory by Henri Cohen provides a comprehensive introduction to the use of computational methods in the study of algebraic number theory. From basic concepts to advanced techniques, the book covers a wide range of topics including factorization, primality testing, class group computation, and more. It is a valuable resource for students and researchers interested in the intersection of number theory and computer science.
A Course in Computational Algebraic Number Theory Review
A Course in Computational Algebraic Number Theory (2000) by Henri Cohen provides detailed insights into the intersection of algebraic number theory and computational methods. Here are three reasons why this book stands out:
- Explains complex theories in clear and accessible language, making it suitable for both beginners and experts in the field.
- Offers practical algorithms and examples that help readers apply theoretical concepts to real-world scenarios, enhancing understanding and learning.
- The book maintains a dynamic pace with engaging exercises that challenge readers and prevent monotony, ensuring an intellectually stimulating read throughout.
Who should read A Course in Computational Algebraic Number Theory?
Graduate students and researchers in mathematics, computer science, or cryptography
Professionals working in the field of number theory or computational algorithms
Individuals with a strong background in mathematics and a keen interest in advanced computational techniques
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A Course in Computational Algebraic Number Theory FAQs
What is the main message of A Course in Computational Algebraic Number Theory?
The main message of A Course in Computational Algebraic Number Theory is to provide a comprehensive understanding of computational methods in algebraic number theory.
How long does it take to read A Course in Computational Algebraic Number Theory?
The estimated reading time for A Course in Computational Algebraic Number Theory is several hours. The Blinkist summary can be read in a fraction of the time.
Is A Course in Computational Algebraic Number Theory a good book? Is it worth reading?
A Course in Computational Algebraic Number Theory is a valuable resource for those interested in the intersection of algebraic number theory and computational techniques, offering practical insights in a concise format.
Who is the author of A Course in Computational Algebraic Number Theory?
The author of A Course in Computational Algebraic Number Theory is Henri Cohen.
What to read after A Course in Computational Algebraic Number Theory?
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