Admissible primes

Definition

Admissible primes are specific prime numbers that satisfy certain conditions in relation to a given number field and its ring of integers. These primes are essential in the context of strong approximation, as they help determine how local properties at these primes relate to global properties of integers in the number field.

5 Must Know Facts For Your Next Test

1. Admissible primes help ensure that the local conditions at these primes align with global conditions, facilitating the application of strong approximation.
2. A prime is considered admissible if it does not divide the discriminant of the number field and meets specific congruence conditions.
3. The set of admissible primes plays a crucial role in determining whether certain algebraic integers can be simultaneously approximated by rational numbers.
4. Admissible primes can affect the structure of the class group and unit group in the context of algebraic number theory.
5. When analyzing quadratic forms or other algebraic structures, admissible primes can lead to significant insights into the solvability of equations.

Review Questions

• How do admissible primes influence the conditions necessary for strong approximation?
  • Admissible primes are critical because they help ensure that local solutions exist at those primes, which directly influences the ability to extend those solutions to global ones. If a prime is not admissible, it may introduce discrepancies between local and global properties, making it impossible to find a solution across the entire number field. Understanding which primes are admissible allows mathematicians to apply the Strong Approximation Theorem effectively.
• Discuss the implications of having a prime that is not admissible when studying number fields.
  • When a prime is not admissible, it can cause issues such as the failure of local-global principles in number theory. Specifically, it may prevent certain algebraic integers from being approximated by rational numbers at that prime. This means that while local conditions may be satisfied, they cannot be extended to a solution in the global context, which limits our understanding of the structure and behavior of numbers within that field. Consequently, it impacts various aspects such as class groups and units.
• Evaluate how the concept of admissible primes relates to both local fields and global fields in algebraic number theory.
  • Admissible primes serve as a bridge between local and global perspectives in algebraic number theory. They allow us to analyze how local behaviors at these specific primes can be translated into conclusions about the entire number field. When we study local fields, we often find that solutions exist only under certain conditions tied to these admissible primes. Thus, understanding this relationship enhances our ability to apply results like the Strong Approximation Theorem and leads to deeper insights into number theoretic properties that are vital in both theoretical exploration and practical applications.

Related terms

Number Field:

A number field is a finite extension of the rational numbers, which contains elements that can be expressed as roots of polynomials with rational coefficients.

Strong Approximation Theorem: The Strong Approximation Theorem is a result in number theory that provides conditions under which solutions to certain equations exist over global fields based on local solutions.

Local Field:

A local field is a complete field with respect to a discrete valuation, which allows for the study of number theoretic problems locally at each prime.