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VIGRE Seminar:Introducing the p-adic numbers

VIGRE Seminar:

Introducing the p-adic numbers

James Stankewicz

University of Georgia

Oct 9, 2007

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Where did they come from?

We begin with some history:

It’s been known for a long time that for any positive integer M , wecan write any non-negative integer n uniquely in “base M ”:

n = m0 + m1M + m2M 2 + · · ·+ mkM k

That is: the mi are unique integers such that 0≤

mi

≤M −

1.

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We begin with some history:

It’s been known for a long time that for any positive integer M , wecan write any non-negative integer n uniquely in “base M ”:

n = m0 + m1M + m2M 2 + · · ·+ mkM k

That is: the mi are unique integers such that 0≤

mi

≤M −

1.

When the notions of “ideals” and “divisors” were introduced byDedekind and Kroenecker in the late 19th century, acorrespondence emerged between prime numbers p and the

elements z − c in C[z].Namely, the prime numbers p generate the prime ideals of Z whilethe linear polynomials generate the prime ideals of C[z].

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The thought among Kroenecker and his students was that just as

expanding a Taylor or Laurent series f (z) = f (c) + a1(z − c) + . . .tells us information about a function near c, perhaps expanding analgebraic number θ as θ = b0 + b1 p + . . . tells us some informationabout θ.One of his students, Kurt Hensel was particularly intrigued by

expanding about primes and began trying to prove things withthese methods.

Theorem(Hensel’s Lemma: Very Weak Version)

If a polynomial F (x) with integer coefficients has a solutionmod p, say α1 ∈ Z>0 and F (α1) is not 0 mod p where F (x) is the formal derivative, then we have a sequence of positive integers {αn} so that F (αn) ≡ 0 mod pn.

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So we get a “Taylor” series expansion for any algebraic number

whose minimal polynomial satisfies some conditions for a givenprime p.

Example

x2 + 1 ≡ (x− 2)(x− 3) mod 5 and 2x ≡ 0 mod 5 if and only if x≡

0 mod 5

So√−1 has base-5 expansion

2 + 1 × 5 + 2 × 52 + . . .

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But we also get negative numbers

−1 = 4 + 4× 5 + 4 × 52 + . . .

And rational numbers whose denominators do not contain p

−1/4 = 1 + 5 + 52 + 53 + . . .

But we can get denominators with p if we allow “Laurent Series at p”

−1/20 = 1/5 + 1 + 5 + 52 + 53 + . . .

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But we also get negative numbers

−1 = 4 + 4× 5 + 4 × 52 + . . .

And rational numbers whose denominators do not contain p

−1/4 = 1 + 5 + 52 + 53 + . . .

But we can get denominators with p if we allow “Laurent Series at p”

−1/20 = 1/5 + 1 + 5 + 52 + 53 + . . .

Notice that all of these series diverge wildly as real numbers.Hensel’s attempts at dealing with convergence were “mostly prettyconfused(and confusing!)”Nonetheless, he did enough work so that Kurschak was able to

formalize his convergence issues by introducing an absolute value.

VIGRE Seminar:Introducing the p adic numbers

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The p-adic absolute value

Definition

An absolute value is a function on a field K , | · | : K → R>0 sothat

1 |a| = 0 if and only if a = 0

2 |ab| = |a||b|3 |a + b| ≤ |a|+ |b|

The p-adic absolute value obeys a stronger condition than property3(the Archimedian inequality),

|a + b| p ≤ max{|a| p, |b| p},

which we call the Non-Archimedian or ultrametric inequality.


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Rather than giving the explicit construction, we illustrate theabsolute value by example:

|1/4|2 = 4, |1/4| p = 1 if p = 2

| −1

| p = 1 for all p

|14/15|2 = 1/2, |14/15|3 = 3, |14/15|5 = 5, |14/15|7 = 1/7

The field of p-adic numbers, which we can view either as the set of “Laurent series in p” or as the completion of Q with respect to the p-adic metric is denoted Q

p. Note: Hensel introduced the p-adic

numbers in 1897 and the metric didn’t appear until 1912. In 1917Ostrowski proved that the p-adic and the real absolute values areall the only nonequivalent ones that can be put on Q.


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Interesting properties of the metric/topology

The ultrametric inequality forces some interesting geometric andtopological conditions on Q p.


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If |x| p = |y| p then |x + y| p = max{|x| p, |y| p}.


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g p





All triangles are isoceles


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g p






|Q p| = pZ ∪ {0} = { pk|k ∈ Z} ∪ {0}


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|Q p| = pZ ∪ {0} = { pk|k ∈ Z} ∪ {0}{x : |x− a| < p

n

} = {x : |x− a| ≤ pn−1

}


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|Q p| = pZ ∪ {0} = { pk|k ∈ Z} ∪ {0}

{x : |x− a| < pn

} = {x : |x− a| ≤ pn−1

}Q p is totally disconnected and locally compact.


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|Q p| = pZ ∪ {0} = { pk|k ∈ Z} ∪ {0}

{x : |x− a| < pn

} = {x : |x− a| ≤ pn−1


n! tends to 0 as a sequence


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|Q p| = pZ ∪ {0} = { pk|k ∈ Z} ∪ {0}

{x : |x− a| < pn

} = {x : |x− a| ≤ pn−1



1/n oscillates in absolute value between 1 and arbitrarily large

numbers


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|Q p| = pZ ∪ {0} = { pk|k ∈ Z} ∪ {0}

{x : |x− a| < pn

} = {x : |x− a| ≤ pn−1



1/n oscillates in absolute value between 1 and arbitrarily large

numbersFor a sequence ai,

i ai converges ⇐⇒ ai → 0

In this setting, Hensel’s Lemma now becomes a powerful tool forfinding roots of polynomials in Z p = {x : |x| ≤ 1} analogous to

Newton’s Method in the real numbers.


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Can we do Calculus?

Can we do calculus on the p-adic numbers?If we look at continuous functions Q p

→R, integration works just

as well on Q p as on R because there is an exact analogue toLesbesgue measure on Q p (translation invariant, inner and outerregular, finite on compact subsets,etc.)Differentiation is far different on Q p. The mean value theorem

does not hold (you have to do something to even state the meanvalue theorem in a reasonable way because Q p is not an orderedfield) and functions can have a derivative which is everywhere zero,but still be nonconstant(therefore we can’t say that two functionswith the same derivative are equal up to a constant and differential

equations also work very differently).Example

f (x) = 1/|x|2 p x = 0

0 x = 0


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Algebra and Number Theory

The effect they had on Algebra and Number Theory was probablygreatest.In Algebra, the p-adic numbers have spawned many generalizationswhich are now areas of study. These include Local Fields, Localrings(Z p ∩Q = Z( p)), Discrete Valuation Rings, etc.In Number Theory, p-adic methods are such a strong tool for

finding rational or integer solutions to polynomial equations theyare practically indispensable.

Theorem(Hasse-Minkowski) A quadratic form with rational coefficients has a solution in Q if and only if it has a solution in R and a solutionin Q p for every p

And more to the point, we can reduce this to checking a finite listof primes.


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The local-global principle

The local-global principle is simply the idea that if you want tolearn about solutions to polynomial equations globally (over Q or anumber field) you should look at local solutions (over R and Q p ortheir analogues in a number field).As we see this works very well when the degree of our polynomialis less than or equal to 2.

Does it work in general?



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The local-global principle is simply the idea that if you want tolearn about solutions to polynomial equations globally (over Q or anumber field) you should look at local solutions (over R and Q p ortheir analogues in a number field).As we see this works very well when the degree of our polynomialis less than or equal to 2.

Does it work in general?No, there are examples of even cubic curves which have localsolutions everywhere but no solutions globally.And right now trying to figure out exactly “how much” thelocal-global principle fails is a busy research area.



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References(In order of use)1) Gouvea, Fernando Q. (1999).“Hensel’s p-adic Numbers: early

history,”www.math.uwo.ca/ srankin/courses/403/2004/hensel2.pdf 2) Gouvea, Fernando Q. (1997). “ p-adic Numbers : AnIntroduction,” 2nd edition, Springer

3) Robert, Alain M. (2000). “A Course in p-adic Analysis,”Springer.4) Gamzon, Adam (2006). “The Hasse-Minkowski Theorem,”http://digitalcommons.uconn.edu/cgi/viewcontent.cgi?article=1017&context=srhonors theses

5) Clark, Pete (2007). “Rational Quadratic Forms and theLocal-Global Principle,”http://math.uga.edu/∼pete/4400rationalqf.pdf

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