P-adic integers

Definition

p-adic integers are elements of the ring of p-adic numbers that are associated with a prime number p. They can be thought of as infinite sequences of digits in base p, where the most significant digit is on the right and the least significant on the left, providing a way to perform arithmetic that extends beyond the traditional integers. This structure allows for unique properties, such as the ability to converge in a different manner compared to real numbers, making them essential in number theory and algebraic geometry.

5 Must Know Facts For Your Next Test

1. p-adic integers can be formally expressed as the set of all sequences of integers a_n such that each a_n is between 0 and p-1, converging in the p-adic metric.
2. The ring of p-adic integers is denoted by Z_p, which is compact and totally disconnected, contrasting with how real numbers behave.
3. In p-adic integers, addition and multiplication are performed differently than with regular integers, particularly due to the way carries work in base p.
4. p-adic integers can be used to study properties of algebraic varieties and solutions to polynomial equations modulo p.
5. An important feature of p-adic integers is that they provide insight into the local-global principles in number theory, helping to bridge relationships between local and global properties.

Review Questions

• How do the arithmetic properties of p-adic integers differ from those of traditional integers?
  • The arithmetic properties of p-adic integers differ significantly from traditional integers mainly due to their base p representation. In p-adic arithmetic, carries behave differently since addition focuses on divisibility by p rather than conventional place value. For example, while adding two numbers might result in a carry affecting multiple digits in base 10, in base p, the carry only affects positions that are multiples of p. This unique behavior leads to distinct convergence properties and manipulations within the field.
• Discuss how Hensel's lemma applies to finding roots of polynomials in the context of p-adic integers.
  • Hensel's lemma provides a powerful tool for finding roots of polynomials in the context of p-adic integers by extending solutions from mod p cases to full p-adic solutions. It states that if a polynomial has a simple root modulo p, then this root can be lifted to a unique root in the ring of p-adic integers. This process allows for the exploration of polynomial behavior in p-adic settings and showcases how local properties can lead to significant insights into global structure.
• Analyze how the structure of p-adic integers influences their applications in number theory and algebraic geometry.
  • The structure of p-adic integers significantly influences their applications by allowing mathematicians to examine problems from different perspectives. Their compactness and total disconnectedness enable researchers to understand local behaviors that are crucial for proving results about rational points on algebraic varieties. Furthermore, their unique valuation properties lead to insights about congruences and diophantine equations, which form fundamental aspects in number theory. This interplay between local solutions in the realm of p-adics and global solutions over rational numbers plays an essential role in developing theories like local-global principles.

Related terms

p-adic numbers: The completion of the rational numbers with respect to the p-adic valuation, forming a field that extends the idea of measuring distance in a way that focuses on divisibility by p.

Valuation: A function that assigns a value to each element of a field that reflects its size or divisibility properties, particularly important in the context of both p-adic and real number systems.

Hensel's lemma: A result in p-adic analysis that provides conditions under which a polynomial equation has solutions in p-adic integers, playing a crucial role in solving equations within this framework.