Number Theory: An Introduction to Mathematics

564 XIII Connections with Number Theory

then evidently

(P,Q)=(Q,P),(P,P)= 2 hˆ(P)≥ 0.

It remains to show that

(P,Q+R)=(P,Q)+(P,R),

and we do this by proving that

hˆ(P+Q+R)=hˆ(P+Q)+hˆ(P+R)+hˆ(Q+R)−hˆ(P)−hˆ(Q)−hˆ(R).

But, by the parallelogram law,

hˆ(P+Q+R+P)+hˆ(Q+R)=hˆ(P+Q+R+P)+hˆ(P+Q+R−P)

= 2 hˆ(P+Q+R)+ 2 hˆ(P)

and

hˆ(P+Q+R+P)+hˆ(Q−R)=hˆ(P+Q+R+P)+hˆ(P+Q−P−R)

= 2 hˆ(P+Q)+ 2 hˆ(P+R).

Subtracting the second relation from the first, we obtain

hˆ(Q+R)−hˆ(Q−R)= 2 hˆ(P+Q+R)+ 2 hˆ(P)− 2 hˆ(P+Q)− 2 hˆ(P+R).

Since, by the parallelogram law again,

hˆ(Q+R)+hˆ(Q−R)= 2 hˆ(Q)+ 2 hˆ(R),

this is equivalent to what we wished to prove.

Proposition 12The abelian group E is finitely generated if, for some integer m> 1 ,
the factor group E/m E is finite.

Proof LetSbe a set of representatives of the cosets of the subgroupmE.SinceSis
finite, by hypothesis, we can chooseC>0sothathˆ(Q)≤Cfor allQ∈S.Theset

S′={Q′∈E:hˆ(Q′)≤C}

containsSand is also finite. We will show that it generatesE.

Let E′be the subgroup ofE generated by the elements ofS′.IfE′ = E,

chooseP∈E\E′so thathˆ(P)is minimal. Then

P=mP 1 +Q 1 for someP 1 ∈EandQ 1 ∈S.

Since

hˆ(P+Q 1 )+hˆ(P−Q 1 )= 2 hˆ(P)+ 2 hˆ(Q 1 ),

it follows that

hˆ(mP 1 )=hˆ(P−Q 1 )≤ 2 hˆ(P)+ 2 C