Theorem of Ostrowski

Definition

The Theorem of Ostrowski states that every non-Archimedean valuation on the field of rational numbers is equivalent to either the p-adic valuation for some prime p or the trivial valuation. This theorem is fundamental in understanding the structure of valuations on fields, especially in the context of p-adic numbers and their applications in number theory.

5 Must Know Facts For Your Next Test

1. The Theorem of Ostrowski provides a complete characterization of non-Archimedean valuations on the rational numbers, showing that they are linked to p-adic numbers.
2. Each p-adic valuation corresponds uniquely to a prime number, allowing for distinct valuations based on different primes.
3. The theorem demonstrates that the only other type of valuation besides p-adic valuations is the trivial valuation, which assigns zero to all non-zero elements.
4. Ostrowski's theorem has implications in number theory, particularly in understanding local-global principles in arithmetic.
5. The theorem highlights the importance of p-adic numbers in modern mathematics, illustrating their role as a critical tool for solving problems in algebraic number theory.

Review Questions

• How does the Theorem of Ostrowski relate to the concept of p-adic valuations?
  • The Theorem of Ostrowski establishes that every non-Archimedean valuation on the rational numbers can be classified as either a p-adic valuation associated with a prime p or as the trivial valuation. This connection is crucial because it shows that p-adic valuations are the only non-trivial valuations available, thereby emphasizing their unique structure and role in number theory.
• Discuss the significance of Ostrowski's theorem in understanding the properties of non-Archimedean fields.
  • Ostrowski's theorem is significant because it provides a comprehensive view of how non-Archimedean fields operate, specifically showing that they are fundamentally connected to p-adic numbers. This relationship allows mathematicians to utilize properties of p-adic valuations when studying various aspects of non-Archimedean fields, leading to deeper insights into their algebraic structures and interactions.
• Evaluate how the Theorem of Ostrowski impacts local-global principles in algebraic number theory.
  • The Theorem of Ostrowski plays a pivotal role in local-global principles within algebraic number theory by clarifying how local properties (as seen through non-Archimedean valuations) relate back to global properties (the behavior over rational numbers). By establishing that all non-Archimedean valuations stem from p-adic valuations or are trivial, it allows mathematicians to better understand when certain properties hold locally at primes and how these conditions can translate into global results across rational numbers.

Related terms

p-adic valuation:

A function that assigns a non-negative integer to each rational number, reflecting how divisible that number is by a given prime p.

Valuation: A function that assigns values to elements of a field, which gives a measure of their size or magnitude, playing a crucial role in the study of algebraic structures.

Non-Archimedean field: A field equipped with a non-Archimedean valuation, where the triangle inequality is replaced by a stronger condition, leading to distinct properties compared to Archimedean fields.