The p-adic Numbers - Delta Epsilons

17 Pages·2009·0.26 MB·English

by  Akhil Mathew

The p-adic Numbers
Akhil Mathew
Math155,ProfessorAlanCandiotti
4 May 2009
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 1/17
The standard absolute value on R: A review
Recall the following properties of the regular absolute value on R:
|·|∞
x 0 with equality iff x = 0
| |∞ ≥
xy = x y , x,y R
| |∞ | |∞| |∞ ∈
x +y x + y (Triangle inequality)
∞ ∞ ∞
| | ≤ | | | |
The standard absolute value induces a notion of distance between two
elements of R, the distance between x,y being
x y .
∞
| − |
Absolute values are studied on more general fields in algebra.
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 2/17
The p-adic valuation on Q
We define the p-adic valuation: If x = 0 is an integer, p a fixed prime, pr
6
the maximum power dividing x,
r
1
x = .
| |p (cid:18)2(cid:19)
If r Q, we have r = a/b for a,b Z, and we set
∈ ∈
a
r = | |p.
| |p b
| |p
This is the p-adic absolute value, defined only on Q. (Also 0 = 0.)
| |p
x 0 with equality iff x = 0
| |p ≥
xy = x y , x,y Q
| |p | |p| |p ∈
x +y max(x , y ) (Non-archimedean inequality: this is
| |p ≤ | |p | |p
stronger than the Triangle Inequality)
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 3/17
p-adic Distance
We can define a new distance and thus a topology on Q from the
valuation : the distance between x,y is
|·|p
x y .
| − |p
x,y are close iff x y is divisible by a high power of p.
−
A sequence a in Q converges p-adically to a if to all (cid:15) > 0, there
n
{ }
exists M such that
n > M implies a a < (cid:15), or lim a a = 0.
| n − |p | n − |p
A sequence a is p-adically Cauchy if to (cid:15) > 0, there is S s.t.
n
{ }
m,n > S a a < (cid:15).
→ | n− m|p
Unlike in R, a p-adically Cauchy sequence need not converge p-adically!
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 4/17
Completions and Q
p
R is the completion (= filling in holes appropriately) of Q w.r.t. the
standard absolute value.
The p-adic numbers Q are the completion of Q w.r.t. the valuation .
p |·|p
Addition, subtraction, multiplication, division extend to the
completion—Q is a field
p
Q Q , just as Q R = Q
p ∞
⊂ ⊂
The absolute value extends to Q by continuity (Q is dense in Q )
|·|p p p
Q is complete with respect to the extended : Any Cauchy
p |·|p
sequence in Q has a limit in Q
p p
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 5/17
Infinite sums in Q
p
Let a be a sequence in Q . We say that ∞ a = a converges to
{ n} p n j=0 j
a ∈ Qp if the partial sums Sn = j=0aj conPverge to a.
P
Theorem
∞
The sum a converges if and only if lima = 0.
j=0 j j
P
Proof.
One implication: straightforward. Suppose a 0; pick (cid:15) > 0 and choose
j
→
N large so that n > N a < (cid:15). Then
→ | n|p
max(m,n)
m,n >N means |Sn −Sm|p = (cid:12)(cid:12) an(cid:12)(cid:12) ≤ mj>aNx|aj|p < (cid:15),
(cid:12)j=minX(m,n)+1 (cid:12)
(cid:12) (cid:12)p
(cid:12) (cid:12)
(cid:12) (cid:12)
so the partial sums are Cauchy and consequently converge.
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 6/17
An example
By substituting x = 2 in the identity 1 = 1+x +x2+x3+..., Euler
1−x
erroneously concluded 1+2+4+ = 1 in R.
··· −
Example
In Q ,
2
1+2+4+8+ = 1.
··· −
Indeed,
n
S = 2j = 2n+1 1,
n
−
Xj=0
so
S ( 1) = (0.5)n+1 0 as n .
| n− − |2 → → ∞
Corollary
Q is not an ordered field.
2
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 7/17
The Heine-Borel Theorem
Theorem (Heine-Borel)
A set in R is compact if it is closed and bounded.
This makes sense for Q too, where point-set topology works similarly.
p
Let A Q . A is open if for x A, there is s > 0 s.t.
p
⊂ ∈
N (x) y : y x < s A;
s ≡{ | − |p }⊂
A is closed if Q A is open. B Q is compact if every open covering
p p
− ⊂
of B has a finite subcovering. C is called bounded if there exists M > 0
such that x C implies x C.
∈ | |p ≤
Notice how similar these notions are to R!
Theorem (p-adic Heine-Borel)
A set in Q is compact if it is closed and bounded.
p
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 8/17
The ring Z
p
We define
Z = x Q : x 1 ;
p { ∈ p | |p ≤ }
this is the analog of the unit interval in R.
Theorem
Z is a ring.
p
Proof.
If x 1, y 1, then xy = x y 1. Also
| |p ≤ | |p ≤ | |p | |p| |p ≤
x +y max(x , y ) 1.
| |p ≤ | |p | |p ≤
Notice how important the nonarchimedean property is.
Now Z Z , and in fact m/n Z if p -n.
p p
⊂ ∈
Theorem
The ideals of Z are of the form prZ for r 0. Z is thus a principal
p p p
≥
ideal domain.
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 9/17
The p-adic expansion
A real number x [0,1] can be represented by a sum b 2−n where
∈ n≥0 n
each bn 0,1 —the binary expansion. For p-adic nuPmbers, the sum
∈{ }
goes in the opposite direction:
Theorem
Any element x Z can be expressed uniquely as an infinite sum
p
∈
x = a pn = a +a p+a p2+a p3+...,
n 0 1 2 3
Xn≥0
where each a = 0,1,..., or p 1.
n
−
For x Q , we have a similar expansion, but we may have a finite number
p
∈
of terms a pn with n < 0.
n
AkhilMathew(Math155,ProfessorAlanCandiotti) Thep-adicNumbers 4May2009 10/17