142 III More on Divisibility
We have already seen in§6 of Chapter II that the ringGof Gaussian integers is
a Euclidean domain, withδ(α)=N(α). We now show that the ringEof Eisenstein
integers is also a Euclidean domain, withδ(α)=N(α).Ifα,β∈Eandα=0, then
βα−^1 =βα′/αα′=r+sρ,wherer,s∈Q. Choosea,b∈Zso that
|r−a|≤ 1 / 2 , |s−b|≤ 1 / 2.Ifκ=a+bρ,thenκ∈Eand
N(βα−^1 −κ)={r−a−(s−b)/ 2 }^2 + 3 {(s−b)/ 2 }^2
≤( 3 / 4 )^2 + 3 ( 1 / 4 )^2 = 3 / 4 < 1.Thusβ−κα∈EandN(β−κα)<N(α).
SinceGandEare Euclidean domains, the divisibility theory of Chapter II is valid
for them. As an application, we prove
Proposition 12The equation x^3 +y^3 =z^3 has no solutions in nonzero integers.
Proof Assume on the contrary that such a solution exists and choose one for which
|xyz|is a minimum. Then(x,y)=(x,z)=(y,z)=1. If 3 did not dividexyz,then
x^3 ,y^3 andz^3 would be congruent to±1 mod 9, which contradictsx^3 +y^3 =z^3 .So,
without loss of generality, we may assume that 3|z.Thenx^3 +y^3 ≡0 mod 3 and,
again without loss of generality, we may assume thatx ≡1 mod 3,y≡−1 mod 3.
This implies that
x^2 −xy+y^2 ≡3mod9.Ifx+yandx^2 −xy+y^2 have a common prime divisorp,thenpdivides 3xy,since
3 xy=(x+y)^2 −(x^2 −xy+y^2 ), and this impliesp=3, since(x,y)=1. Since
(x+y)(x^2 −xy+y^2 )=x^3 +y^3 =z^3 ≡0 mod 27,(^0101)
–1:Gaussian integers –3:Eisenstein integers
i ρ
GOEO
Fig. 1.Gaussian and Eisenstein integers.