Extending the p-adic norm to the field of real numbers

One of the most beautiful problems that I've encountered was the famous Monsky theorem that asserts that one can not tile a rectangle by an odd number of equal area triangles.

This problem is wildly discussed in the forums, yet it is highly sophisticated and requires a wide algebra knowledge.

Among thing required is the fact that one can extend the p-adic absolute from the rational field to the real field, I would like to present this extension here.

I will assume that the reader is familiar with some basic algebra materials that one ought to learn in a second year of undergraduate studies, such as:

Linear algebra,group theory , determinant map, field extension, set theory.

The p-adic norm on the rationals:

As an application of the fundamental theorem of arithmatics every rational number can be written: where .

the p-adic absolute value of is define to be:

basic facts:

1.

2. (ultrametric inequality)

Proof of (2) : assume

then

we have :
. but is an integer, so
altogether , and the proof is finished.
(because of property 1)

actually one can prove

Our goal now is to construct an new absolute value on , such that it will still have properties 1 and 2 (on the real numbers )and Will be an extension of our absolute value on the rationals for all

. .

first step: extending to finite extensions. The Norm

let us assume we have two fields where is a finite extension of .

given an element , considering as a vector space over multiplying by

is a linear transformation! so we can note it as . being a finite dimensional linear transformation

its have a well define determinant. the norm of is defined to be .

practically one can calculate this norm by considering what multiplying by does to basis elements.

Example: let us consider the quadratic extension every element can be written as . let see what happens when multiplying with the basis element

.

writing it as a matrix we have:

which means :

consider the case !

because of the multiplicative nature of the determinant it is not hard to show that :

.

also note .

now we can advance one little step towards our destination.

suppose we have where is a finite extension of , and that somehow

we managed to extend the p-adic valuation on , and we would like to further extend it to

the field , we define it to be:

it is easy to see that this absolute value is still the same if , and still have the multiplicative nature that the absolute value should have.

the hard part is to show that the new absolute value still satisfy the ultrametric inequality, but
i will not give it here, i refer you to (*)

second step: extending the norm to transcendental extension

assume where we managed to define the p-adic absolute value on and we want to extend it to , but this time and is a transcendental element over ,

what can we do?

this time it turns out that we have a lot of freedom. first pick some , it is known that

every element can be uniquely written as

.

we define

.

this is a little complicated but one can check this to be an extension of the p-adic valuation.

so far we have managed to extended the p-adic valuation a 'little further' than we had,

how can we be sure that we can do this procedure for until we reach the whole real numbers

the magic words are Zorn's lemma!

we define a partial order on the set of all p-adic absolute values that are defined on some subfield of the real numbers much similar to how we use Zorn's lemma in the Hahn-Banach Theorem

it is easy to see that every totally ordered subset have an upper bound and that we have a maximal element.

Claim: this maximal element is defined over all the real numbers.

Proof: assume not , we have and it is defined on then one can extend by adding to

an element , can be algebraic or transcendental extension

but we still know how to extend the absolute value on it! contradiction to the maximality.

(*) for further information I advise you to read Cassel's local Fields book chapter 6-7

Remark 1: There is of course lots of ways to extend the norm and the absolute value!

Remark 2: The use of the axiom of choice is seem to be very needed, for example:no one can prove the existence of nontrivial automorphisms of the complex field with out it.

Nice problem:

Try proving that you can not tile a rectangle with odd number of - right triangles

without Monsky Theorem

symmetric strategy in game theory

I will try two new things in this post:

1. I will write seriously about math.

2. Since this post is interesting enough, I will write it in English

Throughout my academic experience, I've managed to meet several

"mathematical games" that have in common the same winning strategy

argument. I will present them blow:

1. Name of game: Pennies on Table

Game laws: your opponent and you put a penny on a round table,

each one in his turn. The pennies must not intersect nor

get out of the boundary of the table.

the one that can't put any more pennies loses the game.

2. Name of game: Determinant Tic-Tac-Toe.

Game laws: your opponent and you have an empty NxN matrix,

your opponent writes 1 on some entry and you write 0 in your

turn. (it does not matter who begins). In the end of the game

you check what the determinant of the matrix is. If it is 0 you

win, else your opponent wins.

3. Name of game: one-dimensional  Tic-Tac-Toe

Game laws: you have a one dimensional array, that is a 1xN matrix.

your opponent and you write 'X' on an entry each in his turn.

whoever write three 'X's in a row win.

I read the third problem in a lovely book by Martin Gardner who was wondering how the games of the two-dimensional 'flatlander' should look like.

(they obviously have to be one-dimensional…)

I will now present the solutions of the problems, so if you want to think about them you should not read the last part.

Pennies on Table:

At first I thought that the radius of the penny and the table is need in order to solve it, but actually you only need to know that the table is larger than the penny….!

The key here is who begin, for the one who begins has the upper hand:

you begin by putting a penny in the exact center. Now, you goal is to make the game radial symmetric: whenever your opponent puts a coin, you put a coin in the exact opposite place related to the center, it is not hard to understand the if the opponent can put his coin then so are you! The game will end in his defeat.

Determinant Tic-Tac-Toe:

I want to leave for you the special cases on N=2,3

The case N=3 is very interesting and was presented to me originally, but is quite simple.

We will call the players 1 and 0.

It is quite surprising but 0 always have a winning strategy, I was surprise because anyone who know enough linear algebra or measure theory know that 'most' of the binary (with 1and 0 entries) matrices are invertible! [1]

For general N you have to split to two cases:

N>3 is even.

if 1 begins then 0 has a winning strategy , all he have to do is to divide the matrix into upper half and lower half. When 1 writes on an entry then 0 write on the mirror entry. For example if N=4 and one writes on the (1,1) then 0 writes 0 on the (1,4) entry…

through this way, when the game is finished:

the sum of row 1+row N=(1,1,…1)

and the sum of row 2+row N-1= (1,1,…,1)

which imply that the rows are linearly dependent and therefore the determinant is 0.

N is even and 0 begins

Amazingly the symmetry strategy is not ruin! Because the opponent is enforced to write 1!

You begin with a 0, even if 1 know about your vicious symmetric plan, he cannot do any thing because he cannot touch the mirror entry with his 1,

So he will have to write 1 in other entry and you carry on by your same strategy…

Again I want to leave for you the interesting case where N is odd, which have a slight twist. (but still a winning strategy for 0!)

One-dimensional  Tic-Tac-Toe:

Again like many similar games the parity of N is important, if N is odd the one who begins  has a strategy  for the win: He first but he's X in the middle of the board whenever the opponent writes X on an entry you put X on the mirror part of the array. It is easy to see that he
will ll reach three row of 'X' s first.

I will leave with the amazing fact ( that was written in the book) :

The one dimensional tic-tac-toe the case when N is even is quite complicated and

there is no obvious winning strategy when N is large enough (such that is promised by Zermelo Theorem) despite the simplicity of the game.

Life is tough for flatlanders!

[1] J. Komlos, On the Determinant of (0,1) Matrices, Studia Scientiarum Mathematicarum Hungarica 2 (1967)

עכשיו שוב שקיעה

"עכשיו שוב שקיעה, אלא אני לבדי בה חותר אל החוף.

נסחף יחידי עם זרמי השמש, עם זרמי המכאוב"

בשורות הנוקבות והמהפנטות הללו של המשורר גבריאל פרייל היה לי קשה להזכר.

קראתי אותן פעם בספר "וזרח השמש ובא השמש" לזכרו של ליאור יונתן ומאז התקשתי לשחזר אותן ואת מיקומו של הספר.

ניסיתי לשווא להעלות אותן מן האינטרנט עם שלל הקומבינציות האפשרויות בעברית באנגלית וביידיש.

הפתיע אותי שנכשלתי, שהרי האינטרנט הוא מקור בלתי נגמר של ידע.

נו טוב, לפחות הפעם אם מישהו יחפש אותן הוא ימצא.

עבר הרבה זמן מאז כתבתי כאן לאחרונה, נראה לי כמו נצח. עבורי, לעיין בפרסומים הישנים שלי מלפני שנתיים או שלוש,

או בתכתובות הישנות שלי ממסנג'ר (כן! מסנג'ר, פעם הייתה תוכנה כזאת) זה כמו לרגל אחרי אדם שונה לחלוטין ולנסות לתהות על קנקנו...

ובכן, אין טעם לדבר עליו מילים לא יפות, כי בדבר אחד הוא לפחות צדק:

זה כיף לכתוב ככה למגירה ולהביע את עצמך באיזה בלוג הזוי!

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