Number Theory: An Introduction to Mathematics

286 VI Hensel’sp-adic Numbers NowE = F(i)containsCand the absolute value onEreduces to the usual absolute value onC. To prove th ...

6 Locally Compact Valued Fields 287 The fieldFis certainly complete if it is locally compact, since any fundamental sequence is ...

288 VI Hensel’sp-adic Numbers ‖a‖ 0 =max 1 ≤i≤n |αi| is a norm onE. Since the fieldFis locally compact, it is also complete. Hen ...

6 Locally Compact Valued Fields 289 thus has prime characteristicp. It follows from Proposition 4that the restriction toQ of the ...

290 VI Hensel’sp-adic Numbers whereπis a generating element for the principal idealM,αn∈Sandαn=0forat most finitely manyn<0. ...

VII The Arithmetic of Quadratic Forms We have already determined the integers which can be represented as a sum of two squares. ...

292 VII The Arithmetic of Quadratic Forms characteristic. For any fieldF, we will denote byF×the multiplicative group of all non ...

1 Quadratic Spaces 293 (ei,ek)= ∑n j,h= 1 τjiβjhτhk, whereβjh=(e′j,e′h), the matrixB=(βjh)is symmetric and A=TtBT. (2) Two symme ...

294 VII The Arithmetic of Quadratic Forms Proposition 1If a quadratic space V contains a vector u such that(u,u)= 0 ,then V=U⊥U ...

1 Quadratic Spaces 295 (i) dimU+dimU⊥=dimV; (ii)U⊥⊥=U; (iii)U⊥is non-singular if and only if U is non-singular. Proof There exis ...

296 VII The Arithmetic of Quadratic Forms Proof SinceVis non-singular, so also areUandW, and sinceVcontains an isotropic vectorv ...

1 Quadratic Spaces 297 Proof LetU+be a maximal positive definite subspace of the quadratic spaceV.Since U+is certainly non-singu ...

298 VII The Arithmetic of Quadratic Forms Proof By choosing an orthogonal basis forVwe are reduced to showing that ifα,β, γ∈F×q, ...

1 Quadratic Spaces 299 Proposition 14Let V be a non-singular quadratic space. If U is a totally isotropic subspace with basis u ...

300 VII The Arithmetic of Quadratic Forms The concept of isometry is only another way of looking at equivalence. For if φ:V →V′i ...

1 Quadratic Spaces 301 Proposition 18Let V be a quadratic space with two orthogonal sum representations V=U⊥W=U′⊥W′. If there ex ...

302 VII The Arithmetic of Quadratic Forms Proof This follows at once from Proposition 18, sinceU′is also non-singular and V=U⊥U⊥ ...

2 The Hilbert Symbol 303 The subspaceHhas twom-dimensional totally isotropic subspacesU 1 ,U 1 ′such that H=U 1 +U 1 ′, U 1 ∩U 1 ...

304 VII The Arithmetic of Quadratic Forms The following lemma shows that the Hilbert symbol can also be defined in an asymmetric ...

2 The Hilbert Symbol 305 Proposition 25Let p be an odd prime and a,b∈Qpwith|a|p=|b|p= 1 .Then (i)(a,b)p= 1 , (ii)(a,pb)p= 1 if a ...