Book contents
- Frontmatter
- Contents
- Introduction
- 1 Cyclohexane, cryptography, codes, and computer algebra
- I Euclid
- II Newton
- III Gauß
- IV Fermat
- V Hilbert
- Appendix
- 25 Fundamental concepts
- Sources of illustrations
- Sources of quotations
- List of algorithms
- List of figures and tables
- References
- List of notation
- Index
- The Holy Qur'ān (732)
25 - Fundamental concepts
from Appendix
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Introduction
- 1 Cyclohexane, cryptography, codes, and computer algebra
- I Euclid
- II Newton
- III Gauß
- IV Fermat
- V Hilbert
- Appendix
- 25 Fundamental concepts
- Sources of illustrations
- Sources of quotations
- List of algorithms
- List of figures and tables
- References
- List of notation
- Index
- The Holy Qur'ān (732)
Summary
This appendix presents some of the basic notions used throughout the text, for the reader's reference. By necessity, this is kept rather short and without proofs; we indicate, however, reference texts where these can be found. The reader is required to either have previous acquaintance with the material or be willing to read up on it. Our presentation is too concise for self-study; its purpose is to fix the language and point the reader to those areas, if any, where she needs brushing up.
The first five sections deal with algebra: groups, rings, polynomials and fields, finite fields, and linear algebra. Then we discuss finite probability spaces. After this mathematical background come some fundamentals from computer science: O-notation and a modicum of complexity theory.
Groups
The material of the first three sections can be found in any basic algebra text, such as Hungerford (1990) or the latest edition of van der Waerden's (1930b, 1931) classic on Modern Algebra.
DEFINITION 25.1. A group is a nonempty set G with a binary operation ·: G × G → G satisfying
◦ Associativity: ∀ a, b, c ∈ G (a · b)· c = a · (b · c),
◦ Identity: ∃1 ∈G ∀a ∈G a ·1 = 1 ·a = a,
◦ Inverse: ∀a ∈G ∃a−1 ∈G a·a−1 = a−1 ·a = 1.
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- Information
- Modern Computer Algebra , pp. 703 - 724Publisher: Cambridge University PressPrint publication year: 2013