Book contents
- Frontmatter
- Dedication
- Contents
- Preface to First Edition
- CHAPTER I Divisibility
- CHAPTER II The Gaussian Primes
- CHAPTER III Polynomials over a field
- CHAPTER IV Algebraic Number Fields
- CHAPTER V Bases
- CHAPTER VI Algebraic Integers and Integral Bases
- CHAPTER VII Arithmetic in Algebraic Number Fields
- CHAPTER VIII The Fundamental Theorem of Ideal Theory
- CHAPTER IX Consequences of the Fundamental Theorem
- CHAPTER X Ideal Classes and Class Numbers
- CHAPTER XI The Fermat Conjecture
- References
- List of Symbols
- Index
CHAPTER III - Polynomials over a field
- Frontmatter
- Dedication
- Contents
- Preface to First Edition
- CHAPTER I Divisibility
- CHAPTER II The Gaussian Primes
- CHAPTER III Polynomials over a field
- CHAPTER IV Algebraic Number Fields
- CHAPTER V Bases
- CHAPTER VI Algebraic Integers and Integral Bases
- CHAPTER VII Arithmetic in Algebraic Number Fields
- CHAPTER VIII The Fundamental Theorem of Ideal Theory
- CHAPTER IX Consequences of the Fundamental Theorem
- CHAPTER X Ideal Classes and Class Numbers
- CHAPTER XI The Fermat Conjecture
- References
- List of Symbols
- Index
Summary
The ring of polynomials. A non-empty set S is called a commutative ring if there are two operations, denoted by + and · such that for all a, b, c ∈ S, (i) a + b = b + a ∈ S, (ii) (a + b) + c = a+ (b + c), (iii) there is a 0 in S such that a + 0 = a for all a in S, (iv) for each a in S there is an element −a in S such that a + (−a) = 0. Further, (v) a · b = b · a ∈ S, (vi) a · (b · c) = (a · b) · c, and (vii) a · (b + c) = a · b + a · c. A set satisfying (i)-(iv) is said to be a commutative group under the operation +. For example, the sets J, G, and H are commutative rings, as are the set of all rational numbers, which we denote by R, and the set of all complex numbers, which we denote by C. Note that the last two examples have a further property: they admit division by non-zero elements.
- Type
- Chapter
- Information
- The Theory of Algebraic Numbers , pp. 25 - 43Publisher: Mathematical Association of AmericaPrint publication year: 1975