What is the p-adic metric?
A -adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime are related to proximity in the so called ” -adic metric.”
How do you calculate p-adic expansion?
The proof of Theorem 3.1 gives an algorithm to compute the p-adic expansion of any rational number in Zp: (1) Assume r < 0. (If r > 0, apply the rest of the algorithm to −r and then negate with (2.2) to get the expansion for r.) (2) If r ∈ Z<0 then write r = −R and pick j ≥ 1 such that R < pj.
What is the point of p-adic numbers?
The p-adic absolute value gives us a new way to measure the distance between two numbers. The p-adic distance between two numbers x and y is the p-adic absolute value of the number x-y. So going back to the 3-adics, that means numbers are closer to each other if they differ by a large power of 3.
Are the P-ADIC integers complete?
The p-adic integers can also be seen as the completion of the integers with respect to a p-adic metric. Let us introduce a p-adic valuation on the integers, which we will extend to Zp.
Is P-ADIC metric complete?
For example, the field of p-adic analysis essentially provides an alternative form of calculus. More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp.
What is ADIC math?
Filters. (1) See AIDC. 1. (mathematics computing) When combined with prefixes derived (usually) from Latin or Greek names for numbers, used to make adjectives meaning “having a certain number of arguments” (said of functions, relations, etc, in mathematics and functions, operators, etc, in computing).
Is the P-ADIC metric complete?
This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility.
Who invented P-ADIC numbers?
mathematician Kurt Hensel
Abstract. The p-adic numbers were invented at the beginning of the twentieth century by the German mathematician Kurt Hensel (1861–1941). The aim was to make the methods of power series expansions, which play such a dominant role in the theory of functions, available to the theory of numbers as well.
What is ADIC?
ADIC – Assistant Director in Charge.
Who invented P-adic numbers?
Is the P ADIC metric complete?
What does ADIC mean in math?
How is the metric space of the p-adic number complete?
This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure that gives the p -adic number systems their power and utility.
Who was the first person to write the p adic number?
p-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer’s earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory.
Can a field of fractions be written as a p adic number?
The ring of p-adic integers has no zero divisors, so we can take the field of fractions to get the field Q p of p-adic numbers. Note that in this field of fractions, every non-integer p-adic number can be uniquely written as p −n u with a natural number n and a unit u in the p-adic integers.
Why are p-adic numbers important to number theory?
The p -adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p -adic analysis essentially provides an alternative form of calculus .