Solving Linear Equations over p –adic Integers By Joseph Molina, Robert Gudino, Dr. Zair Ibragimov

1

0

p –adic Integer expansions

p –adic norms & Geometry

Introduction

The goal above is to get the !! in every term to give us a solution of 0, keeping in mind that (!! = !5). Therefore, we must get it to equal 5 in order for the cycle, the clock, to become complete, then we will carry a one as follows: (We will use figure 1 as a guide).!

2+ !0! = 5! ⇒; !!0! = 3! !1(carried on) +!1+ !!1! = !5 ⇒; !!1! = 3 1(carried on) +!0+ !!2! = 5 ⇒; !!2! = 4 1(carried on) +!0+ !!3! = 5 ⇒; !!3! = 4

10

23

4

Figure 1

!!!!!11 = !1 ∙ 2! + !1 ∙ 2! + !0 ∙ 2! + !1 ∙ 2! + !0 ∙ 2! + !… x!!13! = !1 ∙ 2! + !0 ∙ 2! + !1 ∙ 2! + !1 ∙ 2! + !0 ∙ 2! + !… !!!!!!!!!!!!= !1 ∙ !2! + 0 ∙ 2! !+ 1 ∙ 2! + !1 ∙ 2! + !0 ∙ 2! !!!!!!!!!!!!= !!!!!!!!!!!!!+!1 ∙ !2! + !0 ∙ 2! + !1 ∙ 2! + !1 ∙ 2! + !0 ∙ 2! !!!!!!!!!!!!= !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+!0! ∙ 2! + !0 ∙ 2! + !0 ∙ 2! + !0 ∙ 2! + !0 ∙ 2! +⋯ !!!!!!!!!!!!= !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+!1! ∙ 2! + !0 ∙ 2! + !1 ∙ 2! + !1 ∙ 2! +⋯!!!!!!!!!!!!!= !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+!0! ∙ 2! + !0 ∙ 2! + !0 ∙ 2! +⋯!!!143!!=!1!·!20!+!1!·!21!+!1!·!22!+1!·!23!+!0!·!24!+!0!·!25!+!0!·!26!+!1!·!27!+!…!

After finding the 2- adic expansion of each of the integers we then proceed by multiplying each term in (1) to each one in (2). Then all like terms must be added with respect to each !! . In doing so the technique of carrying must be used, as shown in the previous example. This yields the final solution (3).

The p -adic integers (elements of ℤ!) can be represented as a formal power series

!!!!!

!!!

, where !! ∈ {0, 1, 2, 3, ... , p -1} and p is a positive prime number.

Figure 2

Mod 5

Mod 2

One of the properties that make p-adic integers and p-adic rationals such an important area of research in mathematics is the p-adic metric and the unique properties of metric spaces with the p-adic metric. A metric space, (d,X) is an ordered pair where X is a non empty set and d is a metric on X. The metric d is a distance function that defines the distance between all elements on a set. For example, the usual metric used is the Archimedean metric and is defined as follows. d(a,b) = |a-b|. In other words, distance as we typically use it, is defined as the absolute value of the difference between two points. A metric space must satisfy the following axioms

1) ! !,! ≥ 0 (separation axiom) 2) ! !,! = 0!!""!! = ! (coincidence axiom) 3) ! !,! = !(!, !) (symmetry) 4) ! !, ! ≤ ! !,! + !(!, !) (triangle inequality)

5) ! !, ! ≤ ! !,! + !(!, !) (triangle inequality)

!!!!!! ≡ 1 (mod 3) ∴ !! = 1

!!!!!!!!!!! + !! !≡ 2 (mod 3) !!!!!!→ 1+ !! !≡ 2 (mod 3) !!!!!!!!!∴ !! = 1

!!!!!!!!!!! + !! !≡ 0 (mod 3) !!→ !!!! + 1! ≡ 0 (mod 3) !!!!!!!!!∴ !! = 2

The p-adic metric satisfies all four of the required conditions for a metric space but also satisfies the following condition.

! !, ! ≤ max!{! !,! ,! !, ! }

This is known as the strong triangle nequality. Since the p-adic metric satisfies this condition, it forms a special kind of metric space, known as an ultra metric space.

In#order#to#perform#operations#using#p1adic#integers,#we#need#to#review#modular#arithmetic.#According#to#our#definition,#!! ∈ {1, 2,… , ! − 1 }#or#!! ∈ ℤ/!ℤ.#When#performing#addition#or#multiplication,#there#will#be#cases#where#we#need#to#use#congruence#classes#to#express#numbers#greater#than#(! − 1)#as#an#integer#in#ℤ/!ℤ.##The#best#non#mathematical#analogy#for#this#is#a#standard#clock#which#is#essentially#the#set#of#integers#modulo#12.#Letting#the#modulus#(12),#be#our#zero#point,#the#hour#advances#to#1,#then#2#eventually#to#11#or#(1211).#If#we#advance#another#hour,#we#are#at#our#zero#point#again,#and#then#to#1#rather#than#13.#The#notation#to#express#this#is,#13! ≡ 1!(mod#12).#We#may#express#any#number#this#way.#The#following#are#examples##The#17th#hour#of#a#day#is#congruent#to#5,#or#17! ≡ 5#(mod#12)##The#28th#hour#is#congruent#to#4,#or##28! ≡ 4#(mod#12)##the#351st#hour#is#congruent#to#3,#or#351! ≡ 3#(mod#12)##

Modular Arithmetic

Arithmetic over p–adic Integers !!!!10 = !!!!!1 ∙ 3! !!!!!!+ !!0 ∙ 3! + !!!!!!!!!!!1 ∙ 3! !!!!!!!!!!+ !!!!0 ∙ 3! !!!!!!!!+ 0 ∙ 3! +⋯ + 29 = !!!!!2 ∙ 3! !!!!!!!+ !0 ∙ 3! + !!!!!!!!!!!!0 ∙ 3! !!!!!!!!!+ !!!!1 ∙ 3! !!!!!!!!!+ 0 ∙ 3! +⋯ 39 = 1+ 2 ∙ 3! + 0+ 0 ∙ 3! + 1+ 0 ∙ 3! + 0+ 1 ∙ 3! + 0+ 0 ∙ 3! +⋯ 39 = 0 ∙ 3! + 1 ∙ 3! + 1 ∙ 3! + 1 ∙ 3! + 0 ∙ 3! +⋯

Arithmetic over p–adic integers (continued)

10! = 1 ∙ 3! + !!0 ∙ 3! + !1 ∙ 3! + 0 ∙ 3! + 0 ∙ 3! +⋯ 29! = 2 ∙ 3! + !!0 ∙ 3! + !0 ∙ 3! + 1 ∙ 3! + 0 ∙ 3! +⋯

That leaves us with the new equation (3): 3 − 7! = !3 · 50!+ !3 · 51!+ 4 · 52!+ !4 · 53!+···

Two things can be noted in the solution above, the expansion consist of only positive integers and it is infinite.!

Acknowledgements I would like to thank our mentor and faculty advisor Dr. Zair Ibragimov for his

guidance through this research experience. Next I would like to thank my partner Robert Gudiño, with whom I worked on this research project. I would also like to thank the Cal State Fullerton department of mathematics, the (STEM)² program, and Citrus College, for making this summer research experience possible. This work was funded by the Department of Education grant #P031C110116 (STEM)².

References

Theorem

Model of !!!

[1] Hungerford, Thomas W.. Abstract algebra: an introduction. 2nd ed. Fort Worth: Saunders College Pub., 1997. Print. [2] Katok, Svetlana. P-adic analysis compared with real. Providence, R.I.: American Mathematical Society ;, 2007. Print [3] Rozikov, U. A.. "What are p-Adic Numers?, What are They Used for." Asia Pacific Mathematics Newsletter. N.p., n.d. Web. 4 Aug. 2014. <http://http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf>. [4] Weisstein, Eric W. "p-adic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/p-adicNumber.html

[2]

For every linear equation of the form !" = ! where !, !, ! ∈ ℤ! , there exist a unique solution over ℤ! .

Proof: According to Hungerford’s Theorem 2.8, [3] if p is a positive prime, the equation !" ≡ ! has a unique solution in ℤ/!ℤ. Hence, there is a unique p-adic digit!!!for every !! in the p-adic power series expansion of !. Therefore, every linear equation in ℤ! has a solution over ℤ! .

Kurt Hensel (1861 – 1941) first introduced p –adics in 1897. The p –adic number system, is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the p-adic metric [1]. p-adic numbers have many unique properties and mathematicians have theorized that they may have applications in computer cryptography as well as in quantum mechanics [2].

Our research focuses on p-adic integers ℤ! . More specifically, we examine some of the techniques for performing operations on p -adic integers, using methods from modular arithmetic. We use those methods to solve linear equations over ℤ! , i.e., equations of the form ax = b, where a, b, x are p -adic integers. This raises the question of whether or not all such equations have solutions in ℤ! . The main result of our research shows that every linear equation over ℤ! has a solution and, moreover, we show that the solution is unique. We note that such equations over the usual integers ℤ do not always have solutions.

Definition

Linear Equations over p-adic Integers

(1) !!!!!!!!!4! = !1 · 3! !!!!+ !1 · 3! !!!!+ !0 · 3! !!!+ !0 · 3! !!!!+ !0 · 3!!!!+!. . .!2 !!!!!!!!!!! = !!! · 3! !!+ !!! · 3! !!+ !! · 3! !!+ !!! · 3! !!+ !!! · 3! +⋯! ! !!!!!!!!!!!!!!!!!!!! · 3! !!+ !!! · 3! !!+ !! · 3! !!+ !!! · 3! !!+ !!! · 3! !!+⋯ !!!!!!!!!!!!!!!!!!!!! · 3! !!+ !!! · 3! !!+ !! · 3! !!+ !!! · 3! !!+ !!! · 3! !!+⋯ (3) 7! = 1 · 3! !!!!+ !2 · 3! !!!!!+ !0 · 3! !!!+ !0 · 3! !!!!+ !0 · 3! !!!+ !0 · 3!+!. . .!

In order to solve these kinds of equations, we break it down into smaller modular equations that are easier to solve. The same rules that applied in modular arithmetic, such as carrying still apply.

Ex: Solve for 4x = 7 over 3-adic.

Given the 5-adic expansion of 7, we can now find the expansion of −7. We start by setting up equations (1) and (2), this will help us in finding the expansion.

!!!!! 1 !!!!!!!!!!!!!!!!!!!!!7 = !2 ∙ 5! !+ !!!1 ∙ 5! !+ !!0 ∙ 5! !+ !0 ∙ 5! + !… !!!!! 2 !!!!!!!!!+ !!!−7 = !!! ∙ 5! + !!! ∙ 5! + !!! ∙ 5! + !!! ∙ 5! + !… !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0!!!!!!!!!0!!!!!!!!!!!!!!!!0!!!!!!!!!!!!!!0!!!!!!!!!!!!!!!!0!

[1]

[3]

[4]

Ex: p-adic absolute value or norm: The p-adic norm is defined as |!|! = !!! . Consider the following examples :

8

1

2

3 4

5 6 7

0 9 10

|27|! = ! |0 ∙ 3! + !0 ∙ 3! + !0 ∙ 3! + !1 ∙ 3!|!! = !13! =

19 |13|! = ! |1 ∙ 3! + !1 ∙ 3! + !1 ∙ 3!!|!! = !

13! = 1