Dedekind criterion
Introduction
The Dedekind criterion is a fundamental concept in the field of algebraic number theory, particularly in the study of algebraic number fields and their ring of integers. Named after the German mathematician Dedekind, this criterion provides a necessary and sufficient condition for a prime ideal in a ring of integers to be unramified in an extension field. The criterion is pivotal in understanding the behavior of prime ideals under field extensions and plays a crucial role in the study of ideal theory and Galois theory.
Historical Background
Richard Dedekind, a prominent figure in the development of modern algebra, introduced the criterion in the late 19th century. His work laid the groundwork for the rigorous treatment of algebraic structures, particularly through the introduction of Dedekind domains and the concept of ideals. The Dedekind criterion emerged from his efforts to generalize the unique factorization of integers to more complex algebraic structures.
Mathematical Preliminaries
To fully appreciate the Dedekind criterion, it is essential to understand several key mathematical concepts:
Ring of Integers
The ring of integers in an algebraic number field is the set of all algebraic integers within that field. An algebraic integer is a root of a monic polynomial with coefficients in the integers. The ring of integers forms a commutative ring with unity and is a central object of study in algebraic number theory.
Prime Ideals
A prime ideal in a ring is an ideal such that if the product of two elements is in the ideal, then at least one of those elements must be in the ideal. Prime ideals generalize the notion of prime numbers to more abstract algebraic settings. In the context of algebraic number fields, understanding the behavior of prime ideals under field extensions is crucial for analyzing the structure of the ring of integers.
Field Extensions
A field extension is a pair of fields such that one is a subfield of the other. Field extensions are classified by their degree, which is the dimension of the larger field as a vector space over the smaller field. The Dedekind criterion is particularly concerned with unramified extensions, where the prime ideals do not split or ramify.
The Dedekind Criterion
The Dedekind criterion provides a condition for a prime ideal in a ring of integers to remain unramified in an extension field. Specifically, it states that a prime ideal \( \mathfrak{p} \) in the ring of integers \( \mathcal{O}_K \) of a number field \( K \) is unramified in an extension \( L \) of \( K \) if and only if the discriminant of the extension does not vanish modulo \( \mathfrak{p} \).
Formal Statement
Let \( K \) be a number field with ring of integers \( \mathcal{O}_K \), and let \( L \) be a finite extension of \( K \) with ring of integers \( \mathcal{O}_L \). Let \( \mathfrak{p} \) be a prime ideal in \( \mathcal{O}_K \). The Dedekind criterion states that \( \mathfrak{p} \) is unramified in \( L \) if and only if the discriminant \( \Delta_{L/K} \) is not divisible by \( \mathfrak{p} \).
Proof Outline
The proof of the Dedekind criterion involves several steps:
1. **Reduction to Local Case:** The problem is reduced to considering the local ring \( \mathcal{O}_{K, \mathfrak{p}} \), the localization of \( \mathcal{O}_K \) at \( \mathfrak{p} \).
2. **Integral Basis:** An integral basis for \( \mathcal{O}_L \) over \( \mathcal{O}_K \) is chosen, and the discriminant of this basis is computed.
3. **Discriminant and Ramification:** The relationship between the discriminant and the ramification index of \( \mathfrak{p} \) in \( L \) is established, showing that the discriminant's divisibility by \( \mathfrak{p} \) implies ramification.
4. **Converse Argument:** The converse is shown by demonstrating that if \( \mathfrak{p} \) is ramified, then the discriminant must be divisible by \( \mathfrak{p} \).
Applications
The Dedekind criterion has several important applications in algebraic number theory:
Ideal Factorization
The criterion is instrumental in understanding how prime ideals factor in extensions of number fields. It helps determine whether a prime ideal splits, remains inert, or ramifies in the extension.
Class Field Theory
In class field theory, the Dedekind criterion aids in the classification of abelian extensions of number fields. It provides insights into the behavior of primes in these extensions, which is crucial for understanding the Artin reciprocity law.
Computational Number Theory
In computational number theory, the Dedekind criterion is used to develop algorithms for computing the factorization of ideals in number fields. These algorithms are essential for practical applications, such as cryptography and integer factorization.
Examples
To illustrate the Dedekind criterion, consider the following example:
Example 1: Quadratic Extensions
Let \( K = \mathbb{Q} \) and \( L = \mathbb{Q}(\sqrt{d}) \), where \( d \) is a square-free integer. The ring of integers \( \mathcal{O}_L \) is given by \( \mathbb{Z}[\sqrt{d}] \) if \( d \equiv 2, 3 \pmod{4} \) and \( \mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right] \) if \( d \equiv 1 \pmod{4} \). The discriminant \( \Delta_{L/\mathbb{Q}} \) is \( 4d \) or \( d \) depending on the case. A prime \( p \) is unramified if and only if \( p \nmid \Delta_{L/\mathbb{Q}} \).
Example 2: Cyclotomic Extensions
Consider the cyclotomic field \( \mathbb{Q}(\zeta_n) \), where \( \zeta_n \) is a primitive \( n \)-th root of unity. The discriminant of this extension is related to the Euler's totient function \( \varphi(n) \). A prime \( p \) is unramified in \( \mathbb{Q}(\zeta_n) \) if and only if \( p \nmid n \).
Limitations and Extensions
While the Dedekind criterion is powerful, it has limitations. It applies specifically to unramified extensions and does not directly address the behavior of prime ideals in ramified cases. Extensions of the criterion involve more sophisticated tools, such as ramification theory and valuation theory, to handle these cases.
Conclusion
The Dedekind criterion remains a cornerstone of algebraic number theory, providing deep insights into the behavior of prime ideals in field extensions. Its applications extend across various domains, from theoretical mathematics to practical computational problems. As a testament to Dedekind's legacy, the criterion continues to inspire further research and exploration in the rich landscape of algebraic structures.