2 Quadratic Fields 141N(α)=αα′=r^2 −ds^2.EvidentlyN(α)=N(α′),andN(α)=0 if and only ifα=0. From the relation
(αβ)′=α′β′we obtain
N(αβ)=N(α)N(β).An elementαof the quadratic fieldQ(√
d)is said to be anintegerof this field if it
is a root of a quadratic polynomialt^2 +at+bwith coefficientsa,b∈Z. (Equivalently,
the integers ofQ(
√
d)are the elements which arealgebraic integers.)
It follows from Proposition II.16 thatα∈Qis an integer of the fieldQ(√
d)if and
only ifα∈Z. Suppose now thatα=r+s
√
d,wherer,s∈Qands=0. Thenαis
a root of the quadratic polynomial
f(x)=(x−α)(x−α′)=x^2 − 2 rx+r^2 −ds^2.Moreover, this is the unique monic quadratic polynomial with rational coefficients
which hasαas a root.
Consequently, ifαis an integer ofQ(
√
d), then so also is its conjugateα′and its
normN(α)=r^2 −ds^2 is an ordinary integer.
Proposition 11Let d be a square-free integer and defineωby
ω=√
difd≡ 2 or3mod4,
=(√
d− 1 )/ 2 if d≡1mod4.Thenαis an integer of the quadratic fieldQ(√
d)if and only ifα=a+bωfor
some a,b∈Z.
Proof Supposeα=r+s
√
d,wherer,s∈Q. As we have seen, ifs=0thenαis an
integer ofQ(
√
d)if and only ifr∈Z.Ifs=0, thenαis an integer ofQ(√
d)if and
only ifa= 2 randb=r^2 −ds^2 are ordinary integers. Ifais even, i.e. ifr∈Z,then
b∈Zif and only ifds^2 ∈Zand hence, sincedis square-free, if and only ifs∈Z.
Ifais odd, thena^2 ≡1 mod 4 and henceb∈Zif and only if 4ds^2 ≡1 mod 4. Since
dis square-free, this implies that 2s∈Z,s∈/Z. Hence 2sis odd andd≡1 mod 4.
Conversely, if 2rand 2sare odd integers andd≡1 mod 4, thenr^2 −ds^2 ∈Z.The
result follows.
Sinceω^2 =−ω+(d− 1 )/4inthecased≡1 mod 4, it follows directly from
Proposition 11 that the setOdof all integers of the fieldQ(
√
d)is closed under sub-
traction and multiplication and consequently is a ring. In factOdis an integral domain,
sinceOd⊆Q(
√
d).
For example,O− 1 =Gis the ring of Gaussian integersa+bi,wherea,b∈Z.
They form a square ‘lattice’ in the complex plane. SimilarlyO− 3 =Eis the ring of all
complex numbersa+bρ,wherea,b∈Zandρ=(i
√
3 − 1 )/2 is a cube root of unity.
TheseEisenstein integerswere studied by Eisenstein (1844). They form a hexagonal
‘lattice’ in the complex plane.