Number Theory: An Introduction to Mathematics

306 VII The Arithmetic of Quadratic Forms wherea 0 ,b 0 ∈{ 3 , 5 , 7 }anda′,b′∈Z 2. Hence 1 =ax^2 +by^2 = 22 α(a 0 +b 0 y 02 + 8 ...

2 The Hilbert Symbol 307 Table 1.Values of the Hilbert symbol(a,b)FforF=Qv Q∞=RQp:podd a\b 1 − 1 a\b 1 prpr 1 ++ 1 ++++ − 1 +− p ...

308 VII The Arithmetic of Quadratic Forms then ζ 12 −aζ 22 −bζ 32 +abζ 42 =(ξ 12 −aξ 22 −bξ 32 +abξ 42 )(η^21 −aη^22 −bη^23 +abη ...

2 The Hilbert Symbol 309 witha,b,c ∈ F×. Evidentlyaandcare not squares, and ifdis represented by ξ 32 −cξ 42 ,thenbdis not repre ...

310 VII The Arithmetic of Quadratic Forms then(−a,−b)F =− 1 =(− 1 ,−b)Fand(−a,b)F = 1 = (− 1 ,b)F. Thus, for allc∈F×,(−a,c)F=(− ...

2 The Hilbert Symbol 311 a 12 (u 1 ,u 1 )+···+a^2 p(up,up)=(u′ 1 ,u′ 1 )= 0 , and each summand on the left is nonzero. If the s ...

312 VII The Arithmetic of Quadratic Forms where(a,ad)F = (b,bd)F,thenf is equivalent tog. The hypothesis implies (−d,a)F=(−d,b)F ...

3 The Hasse–Minkowski Theorem 313 was inspired. The assumption was first proved by Dirichlet (1837) and will be referred to here ...

314 VII The Arithmetic of Quadratic Forms ∏ v (− 1 ,p)v=(− 1 ,p)p(− 1 ,p) 2 =(− 1 /p)(− 1 )(p−^1 )/^2 ; ∏ v ( 2 ,p)v=( 2 ,p)p( 2 ...

3 The Hasse–Minkowski Theorem 315 We are going to show that there exists an integercsuch thatc^2 ≡amodb. Since±bis a product of ...

316 VII The Arithmetic of Quadratic Forms By the Chinese remainder theorem (Corollary II.38), the simultaneous congruences c≡b 2 ...

3 The Hasse–Minkowski Theorem 317 Proof Only the sufficiency of the condition requires proof. But if the rational quadratic form ...

318 VII The Arithmetic of Quadratic Forms F=K(t)of rational functions in one variable with coefficients from a fieldK,the weak H ...

3 The Hasse–Minkowski Theorem 319 Let (x,y)={f(x+y)−f(x)−f(y)}/ 2 be the symmetric bilinear form associated withf,sothatf(x)=(x, ...

320 VII The Arithmetic of Quadratic Forms thenD:=BtBandE:=BtJBare diagonal matrices: D=diag[d 1 ,...,dn− 1 ,n], E=diag[0,..., 0 ...

3 The Hasse–Minkowski Theorem 321 aξ^2 +(− 1 )(n−^2 )/^2 aη^2 −ζ^2 , is isotropic inQ. Thus it is certainly satisfied ifn ≡0 mod ...

322 VII The Arithmetic of Quadratic Forms 4 Supplements It was shown in the proof of Proposition 41 that if an integer can be re ...

4 Supplements 323 where f 0 (t),f 1 (t),...,fn(t)∈ F[t]. Assume that f 0 does not divide fjfor some j∈{ 1 ,...,n}.Thend:=degf 0 ...

324 VII The Arithmetic of Quadratic Forms where f 1 (x),...,fs(x) ∈ R(x). The question was answered affirmatively by Artin (1927 ...

6 Selected References 325 not so ‘clean’. However, Conway [6] has given an elementary approach to theequiva- lenceof quadratic f ...