Number Theory: An Introduction to Mathematics

6 Sums of Squares 121

Evidentlyγ ∈H if and only if it can be written in the formγ=a 0 h+a 1 i+
a 2 j+a 3 k,wherea 0 ,a 1 ,a 2 ,a 3 ∈Zandh=( 1 +i+j+k)/2. It is obvious that the
product ofhwithi,jorkis again inHand it is easily verified thath^2 =h−1. It
follows thatHis closed under multiplication and hence is a ring.
Define thenormof a quaternionγ=a+bi+cj+dkto be
N(γ)=γγ ̄=a^2 +b^2 +c^2 +d^2.

ThenN(γ)≥0, with equality if and only ifγ=0. Moreover, sinceγ 1 γ 2 = ̄γ 2 γ ̄ 1 ,

N(γ 1 γ 2 )=γ 1 γ 2 γ ̄ 2 γ ̄ 1 =γ 1 γ ̄ 1 γ 2 γ ̄ 2 =N(γ 1 )N(γ 2 ).

Ifγ ∈H,thenN(γ)=γγ ̄ ∈H and henceN(γ)is an ordinary integer. Further-
more,γis a unit inH,i.e.γdivides 1 inH, if and only ifN(γ)=1.
We now show that a Euclidean algorithm may be defined onH. Supposeα,β∈
Handα=0. Then

βα−^1 =r 0 +r 1 i+r 2 j+r 3 k,

wherer 0 ,r 1 ,r 2 ,r 3 ∈Q.Ifκ=a 0 h+a 1 i+a 2 j+a 3 k,then

βα−^1 −κ=(r 0 −a 0 / 2 )+(r 1 −a 0 / 2 −a 1 )i+(r 2 −a 0 / 2 −a 2 )j

+(r 3 −a 0 / 2 −a 3 )k.

We can choosea 0 ∈Zso that| 2 r 0 −a 0 |≤ 1 /2 and then chooseav ∈Zso that
|rv−a 0 / 2 −av|≤ 1 / 2 (v= 1 , 2 , 3 ).Thenκ∈Hand

N(βα−^1 −κ)≤ 1 / 16 + 3 / 4 = 13 / 16 < 1.

Thus if we setρ=β−κα,thenρ∈Hand

N(ρ)=N(βα−^1 −κ)N(α) <N(α).

By repeating this division process finitely many times we see that anyα,β∈H
have agreatest common right divisorδ=(α,β)r. Furthermore, there is aleft B ́ezout
identity:δ=ξα+ηβfor someξ,η∈H.
If a positive integernis a sum of four squares, sayn=a^2 +b^2 +c^2 +d^2 ,then
n=γγ ̄,whereγ=a+bi+cj+dk∈H. Since the norm of a product is the product
of the norms, it follows that any product of sums of four squares is again a sum of four
squares. Hence to prove the proposition we need only show that any primepis a sum
of four squares.
We show first that there exist integersa,bsuch thata^2 +b^2 ≡−1modp.This
follows from the illustration given for Proposition 34, but we will give a direct proof.
Ifp=2, we can takea=1,b=0. Ifp≡1 mod 4 then, by Corollary 29, there
exists an integerasuch thata^2 ≡−1modpand we can takeb=0. Suppose now that
p≡3 mod 4. Letcbe the least positive quadratic non-residue ofp.Thenc≥2and
c−1 is a quadratic residue ofp. On the other hand,−1 is a quadratic non-residue of
p, by Corollary 29. Hence, by Proposition 28,−cis a quadratic residue. Thus there
exist integersa,bsuch that

a^2 ≡−c,b^2 ≡c−1modp,

and thena^2 +b^2 ≡−1modp.