Construction Of Qp With Two Approaches

Construction of Qp with Two ApproachesLanqi FeiUniversity of Marylandlanqifei@terpmail.umd.eduLanqi Fei (UMD)Construction of P-adic Numbers1 / 29

Why do we study P-adic Numbers?The p-adic numbers is a larger number system containing Q, withnicer properties.Lanqi Fei (UMD)Construction of P-adic Numbers2 / 29

Why do we study P-adic Numbers?The p-adic numbers is a larger number system containing Q, withnicer properties.When the p-adic numbers were introduced they considered as anexotic part of pure mathematics without any application.Lanqi Fei (UMD)Construction of P-adic Numbers2 / 29

Why do we study P-adic Numbers?The p-adic numbers is a larger number system containing Q, withnicer properties.When the p-adic numbers were introduced they considered as anexotic part of pure mathematics without any application.It turns out later to have powerful applications in fields like numbertheory, including, for example, in the famous proof of Fermat’s LastTheorem by Andrew Wiles.Lanqi Fei (UMD)Construction of P-adic Numbers2 / 29

Why do we study P-adic Numbers?The p-adic numbers is a larger number system containing Q, withnicer properties.When the p-adic numbers were introduced they considered as anexotic part of pure mathematics without any application.It turns out later to have powerful applications in fields like numbertheory, including, for example, in the famous proof of Fermat’s LastTheorem by Andrew Wiles.Since 80th p-adic numbers are used in applications to quantumphysics.Lanqi Fei (UMD)Construction of P-adic Numbers2 / 29

Overview1Algebraic Construction2Topological Construction3Connecting the Two ConstructionsLanqi Fei (UMD)Construction of P-adic Numbers3 / 29

Algebraic ConstructionLanqi Fei (UMD)Construction of P-adic Numbers4 / 29

P-adic IntegerGiven a prime p, for each integer m, we can write it in base p in a uniqueway,m a0 a1 p a2 p 2 · · · an p n , 0 ai pExample7 1 1 · 2 1 · 22Lanqi Fei (UMD)Construction of P-adic Numbers5 / 29

P-adic IntegerDefinition (P-adic Integer)Let p be a prime. The set of p-adic integers is defined asZp a0 a1 p a2 p 2 . . .where 0 ai pExample1 1 · 2 1 · 22 · · · 1 · 2n · · · 2 Z 2Lanqi Fei (UMD)Construction of P-adic Numbers6 / 29

P-adic Integera0 a1 p a2 p 2 . . .#mod p n[a0 a1 p · · · an1pn 1] 2 Z/p n Zwhere 0 ai pLanqi Fei (UMD)Construction of P-adic Numbers7 / 29

P-adic IntegerThis defines a map from Zp to1Xi 0Q1n 1 Z/pnZai p i 7 ! ([a0 ], [a0 a1 p], . . . , [Lanqi Fei (UMD)n 1Xa1 p i ], [i 0Construction of P-adic NumbersnXa1 p i ], . . . )i 08 / 29

P-adic IntegerThis defines a map from Zp to1Xi 0Q1n 1 Z/pnZai p i 7 ! ([a0 ], [a0 a1 p], . . . , [n 1Xa1 p i ], [i 0nXa1 p i ], . . . )i 0Moreover, we have[nXi 0Lanqi Fei (UMD)ia1 p ]7mod p n1![n 1XConstruction of P-adic Numbersa1 p i ]i 08 / 29

Inverse LimitDefinitionnlim Z/p Z (xn )n2N 2Lanqi Fei (UMD)1Yn 1Z/p n Z xn 7! xnConstruction of P-adic Numbers1, n 1, 2, . . .9 / 29

Inverse LimitTheoremAssociating to every p-adic integer a equivalence classesxn n 1Xai p ii 0P1i 0 ai pithe sequence (xn )n2N ofmod p n 2 Z/p n Z,yields a bijectionZp ! lim Z/p n Z.Example1 2 22 · · · 2n . . .Lanqi Fei (UMD)! ([1], [1 2], [1 2 22 ], . . . ) (1 mod 2, 3Construction of P-adic Numbersmod 4, . . . )10 / 29

P-adic NumbersDefinitionwe extend the domain of p-adic integers into that of the formal series1Xv mav p v ampm · · · a0 a1 p . . . ,where m 2 Z and 0 av p. We call such series p-adic numbers anddenote the set of p-adic numbers as Qp .Lanqi Fei (UMD)Construction of P-adic Numbers11 / 29

Topological ConstructionLanqi Fei (UMD)Construction of P-adic Numbers12 / 29

MotivationR completion of Q with respect to the usual absolute value , whichhas the following properties123 a 0 , a 0 ab a b a b a b We’ll construct p-adic numbers in a similar way, with a di erent absolutevalue.Lanqi Fei (UMD)Construction of P-adic Numbers13 / 29

P-adic Absolute ValueDefinition (P-adic Absolute Value)Let p be a prime. Given a non-zero rational x can write it as follows,a0x p vp (x) 0b00such that p 6 a and p 6 b .mn,where m, n 2 Z,weThe p-adic absolute value is defined as follows, x p pvp (x)and we define 0 p 0.Lanqi Fei (UMD)Construction of P-adic Numbers14 / 29

P-adic Absolute ValueExample125 53 125 5 53 50 33 3 5 50 1 125 5 3 5 !Lanqi Fei (UMD)Construction of P-adic Numbers15 / 29

Absolute Values on QTheorem (Ostrowski’s)Every non-trivial absolute value on Q is either p for some prime p or theusual absolute value .Lanqi Fei (UMD)Construction of P-adic Numbers16 / 29

TopologyIn (Q, d), d(x, y ) xy pAll triangles are isosceles.Any point of ball B(a, r ) {x 2 Q : xa p r } is center.Two balls are either disjoint, or one is contained in the other.Lanqi Fei (UMD)Construction of P-adic Numbers17 / 29

CompletionDefinitionC Cauchy Sequences in Q w.r.t p {(c1 , c2 , . . . )}m Nullsequences in Q {(x1 , x2 , . . . ) xn p ! 0}Lanqi Fei (UMD)Construction of P-adic Numbers18 / 29

CompletionDefinitionC Cauchy Sequences in Q w.r.t p {(c1 , c2 , . . . )}m Nullsequences in Q {(x1 , x2 , . . . ) xn p ! 0}TheoremC forms a ring, and m forms a maximal ideal of C.Lanqi Fei (UMD)Construction of P-adic Numbers18 / 29

CompletionDefinitionWe define the field of p-adic numbers to beQp C/mLanqi Fei (UMD)Construction of P-adic Numbers19 / 29

CompletionDefinitionWe define the field of p-adic numbers to beQp C/mWe extend the p-adic absolute value to Qp by setting x p (x1 , x2 , . . . ) m p lim xn pn!1Lanqi Fei (UMD)Construction of P-adic Numbers19 / 29

CompletionTheoremThe field Qp of p-adic numbers is complete with respect to the absolutevalue p , i.e., every Cauchy sequence in Qp converges with respect to p .Lanqi Fei (UMD)Construction of P-adic Numbers20 / 29

P-adic IntegersDefinitionThe set of p-adic integers is defined asZp : x 2 Qp x p 1is a subring of Qp . It is the closure with respect to p of the ring Z Qp .Lanqi Fei (UMD)Construction of P-adic Numbers21 / 29

P-adic IntegersTheoremThe non-zero ideals of the ring Zp are the principal idealsp n Zp x 2 Qp x p with n1pn0, and we haveZp /p n Zp Z/p n ZLanqi Fei (UMD)Construction of P-adic Numbers22 / 29

IsomorphismTheorem (Cont.)Zp /p n Zp Z/p n Z[x] [a]where a 2 Z satisfies xLanqi Fei (UMD)a p 1pn ,and [a] 2 Z/p n Z is unique.Construction of P-adic Numbers23 / 29

Connecting the Two ConstructionsLanqi Fei (UMD)Construction of P-adic Numbers24 / 29

Connecting Two ApproachesFor each n, we get a homomorphismLanqi Fei (UMD)Zp!x7 !Zp /p n Zp Z/p n Z[x]![an ]Construction of P-adic Numbers25 / 29

Connecting Two ApproachesFor each n, we get a homomorphismZp!x7 !Zp /p n Zp Z/p n Z[x]![an ]Combine the homomorphisms for all n, we get a homomorphismZp !1YZ/p n Zn 1In fact, the we getZp ! lim Z/p n ZLanqi Fei (UMD)Construction of P-adic Numbers25 / 29

Connecting Two ApproachesTheoremThe homomorphismZp ! lim Z/p n Zis an isomorphism (and even homeomorphism).LHS Topological definition of p-adic integersRHS Algebraic definition of p-adic integersLanqi Fei (UMD)Construction of P-adic Numbers26 / 29

Connecting Two ApproachesFor the algebraic side, we define Qp to be the quotient field of p-adicintegers; for the topological side, we can prove Qp quotient field of Zp .Because the two rings are isomorphic, their quotient fields are isomorphic,so two definitions of p-adic numbers coincide.Lanqi Fei (UMD)Construction of P-adic Numbers27 / 29

ReferencesFernando Q. Gouvea (1997)p-adic Numbers: An IntroductionJurgen Neukirch (1999)Algebraic Number TheoryU. A. Rozikov (2013)What are p-adic Numbers? What are They Used for?Lanqi Fei (UMD)Construction of P-adic Numbers28 / 29

Thank YouLanqi Fei (UMD)Construction of P-adic Numbers29 / 29

theory, including, for example, in the famous proof of Fermat’s Last Theorem by Andrew Wiles. Since 80th p-adic numbers are used in applications to quantum physics. Lanqi Fei (UMD) Construction of P-adic Numbers 2/29