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