Number Theory: An Introduction to Mathematics

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^24 ).

It follows thatGa,bis a subgroup ofF×. 

Proposition 30A field F is a Hilbert field if and only if one of the following mutually

exclusive conditions is satisfied:

(A)F is an ordered field and every positive element of F is a square;
(B)there exists, up to equivalence, one and only one anisotropic quaternary quadratic
form over F.

Proof Suppose first that the fieldFis of type (A). Then−1 is not a square, since

− 1 + 1 =0 and any nonzero square is positive. By Proposition 10, any anisotropic

binary quadratic form is equivalent overFto exactly one of the formsξ^2 +η^2 ,−ξ^2 −η^2

and thereforeFis a Hilbert field. Since the quadratic formsξ 12 +ξ 22 +ξ 32 +ξ 42 and

−ξ 12 −ξ 22 −ξ 32 −ξ 42 are anisotropic and inequivalent, the fieldFis not of type (B).

Suppose next that the fieldFis of type (B). The anisotropic quaternary quadratic

form must be universal, since it is equivalent to any nonzero scalar multiple. Hence,

for anya∈F×there exists an anisotropic diagonal form

−aξ 12 −b′ξ 22 −c′ξ 32 −d′ξ 42 ,

whereb′,c′,d′∈ F×. In particular, fora=−1, this shows that not every element

ofF×is a square. The ternary quadratic formh=−b′ξ 22 −c′ξ 32 −d′ξ 42 is certainly

anisotropic. Ifhdoes not represent 1, the quaternary quadratic form−ξ 12 +his also

anisotropic and hence, by Witt’s cancellation theorem,amust be a square. Conse-

quently, ifa∈F×is not a square, then there exists an anisotropic form

−aξ 12 +ξ 22 −bξ 32 −cξ 42.

Thus for anya ∈ F×which is not a square, there exists b ∈ F×such that

(a,b)F=−1. If(a,b)F=(a,b′)F=−1 then, by Lemma 29, the forms

ξ 12 −aξ 22 −b(ξ 32 −aξ 42 ),ξ 12 −aξ 22 −b′(ξ 32 −aξ 42 )

are anisotropic and thus equivalent. It follows from Witt’s cancellation theorem that

the binary formsb(ξ 32 −aξ 42 )andb′(ξ 32 −aξ 42 )are equivalent. Consequentlyξ 32 −aξ 42

representsbb′and(a,bb′)F=1. ThusGahas index 2 inF×for anya∈F×which

is not a square, andFis a Hilbert field.

Suppose now thatFis a Hilbert field. Then there existsa∈F×which is not a

square and, for any sucha, there existsb∈F×such that(a,b)F=−1. Consequently,

by Lemma 29, the quaternary quadratic formfa,bis anisotropic and represents 1. Con-

versely, any anisotropic quaternary quadratic form which represents 1 is equivalent to

some form

g=ξ 12 −aξ 22 −b(ξ 32 −cξ 42 )