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## Über dieses Buch

The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic.

Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform.

The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.

## Inhaltsverzeichnis

### Chapter 1. Unique Factorisation in the Natural Numbers

Abstract
We are so used to working with the natural numbers from infancy onwards that we take it for granted that natural numbers may be factorised uniquely into prime numbers. For example, $$360=2^{3}3^{2}5$$ is the prime factorisation of 360. However, we should notice that there are already senses in which this factorisation is not really unique; we can write $$360=2\times 3\times 5\times 2\times 3\times 2$$, or even $$360=(-2)\times 5\times 3\times (-3)\times 2\times 2$$. Nevertheless, we can see that all these factorisations are “essentially the same”, in a way which we could make precise, and we will do so later.
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### Chapter 2. Number Fields

Abstract
We’ve just seen examples where questions about integers were naturally treated by working in the slightly bigger set $$\mathbb {Z}[i]$$ of Gaussian integersGauss, Carl Friedrich!Gaussian integers. In this chapter we begin the development of some more general theory.
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### Chapter 3. Fields, Discriminants and Integral Bases

Abstract
Discriminant Integral basisBy definition, every number field $$K$$ is a finite extension of $$\mathbb {Q}$$. In particular, if $$K$$ has degree $$n$$, then there must be elements $$\alpha _1,\ldots ,\alpha _n\in K$$ such that every element of $$K$$ can be written as a linear combination
$$x_1\alpha _1+x_2\alpha _2+\cdots +x_n\alpha _n$$
where $$x_1,\ldots ,x_n\in \mathbb {Q}$$.
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### Chapter 4. Ideals

Abstract
In Chap. 2, we saw that the set of integers $$\mathbb {Z}_K$$ in an algebraic number field $$K$$ forms a ring. In this chapter and the next, we are going to begin the study of primes.
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### Chapter 5. Prime Ideals and Unique Factorisation

Abstract
We have already studied unique factorisation in $$\mathbb {Z}$$, and seen how it fails in certain rings of integers of number fields. We have also seen the suggestion that non-uniqueness of factorisation may be remedied by working with ideals. In order to show that this procedure will work generally, we will need to have some concept of what it means for an ideal to be prime.
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### Chapter 6. Imaginary Quadratic Fields

Abstract
It won’t be a surpriseQuadratic field!imaginary|( that fields of low degree over $$\mathbb {Q}$$ are going to be the easiest cases to understand.
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### Chapter 7. Lattices and Geometrical Methods

Abstract
In this chapter, we will prove two fundamental results in algebraic number theory: the finiteness of the class group, and Dirichlet’s Unit TheoremDirichlet, Peter Gustav Lejeune!Dirichlet’s unit theorem, which gives the structure of the group of units in the rings of integers of number fields.
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### Chapter 8. Other Fields of Small Degree

Abstract
The results of Chap. 6 give a fairly complete description of imaginary quadratic fields. But other fields have some different properties, and we will meet some of these for the first time in this chapter.
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### Chapter 9. Cyclotomic Fields and the Fermat Equation

Abstract
Cyclotomic fieldsCyclotomic field are the number fields generated over $$\mathbb {Q}$$ by roots of unityRoot of unity. They played (and still play) an important role in developing modern algebraic number theory, most notably because of their connection with Fermat’s Last TheoremFermat, Pierre de!Fermat’s Last Theorem (see Sect. 9.4). Whole books have been written about cyclotomic fields, but we will just begin to develop a few of their properties.
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### Chapter 10. Analytic Methods

Abstract
Although this is a textbook on algebraic number theory, some interesting algebraic results can be obtained by incorporating some analytic techniques.
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### Chapter 11. The Number Field Sieve

Abstract
Like most pure mathematics, number theory developed with little thought to applications.
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### Backmatter

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