Number Theory: An Introduction to Mathematics

2 Quadratic Fields 143

it follows that

x+y= 9 a^3 ,

x^2 −xy+y^2 = 3 b^3 ,

wherea,b∈Zand 3
b.
We now shift operations to the Euclidean domainEof Eisenstein integers. We have

x^2 −xy+y^2 =(x+yρ)(x+yρ^2 ),

whereρ=(i

√

3 − 1 )/2 is a cube root of unity. Putλ= 1 −ρ,sothat( 1 +ρ)λ^2 =3.
Thenλis a common divisor ofx+yρandx+yρ^2 ,since

x+yρ=x+y−yλ,

x+yρ^2 =x− 2 y+yλ

andx+y≡ 0 ≡x− 2 ymod 3. In factλis the greatest common divisor ofx+yρ
andx+yρ^2 since, for allm,n∈Z,

(m+n+nρ)(x+yρ^2 )−(n+mρ+nρ)(x+yρ)=(mx+ny)λ

and we can choosem,nso thatmx+ny=1. Sinceλ^2 =− 3 ρand sinceρis a unit,
from(x+yρ)(x+yρ^2 )= 3 b^3 and the unique factorization ofbinE, we now obtain

x+yρ=ελ(c+dρ)^3 ,

wherec,d∈Zandεis a unit. From

(x+yρ)/λ=x−λ(x+y)/ 3 =x− 3 a^3 λ

and

(c+dρ)^3 =c^3 − 3 cd^2 +d^3 + 3 cd(c−d)ρ,

by reducing mod 3 we get

ε(c^3 +d^3 )≡1mod3.

Since the units inEare± 1 ,±ρ,±ρ^2 (by the following Proposition 13), this implies
ε=±1. In fact we may supposeε=1, by changing the signs ofcandd. Equating
coefficients ofρ,wenowget

a^3 =cd(c−d).

But(c,d)=1, since(x,y)=1, and hence also(c,c−d)=(d,c−d)=1. It follows
thatc=z^31 ,d=y^31 ,c−d=x^31 for somex 1 ,y 1 ,z 1 ∈Z. Thusx 13 +y^31 =z^31 and

|x 1 y 1 z 1 |=|a|=|z/ 3 b|<|xyz|.

But this contradicts the definition ofx,y,z.