Number Theory: An Introduction to Mathematics

2 Lattices 333

more general definition and, with this warning, believe it will be clear from the context
when this occurs.
The basis of a lattice is not uniquely determined. In facty 1 ,...,ynis also a basis if

yj=

∑n

k= 1

αjkxk (j= 1 ,...,n),

whereA=(αjk)is ann×nmatrix of integers such that detA=±1, sinceA−^1 is
then also a matrix of integers. Moreover, every basisy 1 ,...,ynis obtained in this way.
For if

yj=

∑n

k= 1

αjkxk, xi=

∑n

j= 1

βijyj,(i,j= 1 ,...,n),

whereA=(αjk)andB=(βij)aren×nmatrices of integers, thenBA=Iand hence
(detB)(detA)=1. Since detAand detBare integers, it follows that detA=±1.
Letx 1 ,...,xnbe a basis for a latticeΛ⊆Rn.If

xk=

∑n

j= 1

γjkej (k= 1 ,...,n),

wheree 1 ,...,enis the canonical basis forRnthen, in terms of the nonsingular matrix
T=(γjk), the latticeΛis just the set of all vectorsTzwithz∈Zn. The absolute
value of the determinant of the matrixTdoes not depend on the choice of basis. For if
x′ 1 ,...,x′nis any other basis, then

xi′=

∑n

j= 1

αijxj (i= 1 ,...,n),

whereA=(αij)is ann×nmatrix of integers with detA=±1. Thus

x′k=

∑n

j= 1

γjk′ej (k= 1 ,...,n),

whereT′=(γjk′)satisfiesT′=TAtand hence

|detT′|=|detT|.

The uniquely determined quantity|detT|will be called thedeterminantof the lattice
Λand denoted by d(Λ). (Some authors, e.g. Conway and Sloane [14], call|detT|^2
the determinant ofΛ, but others prefer to call this thediscriminantofΛ.)
The determinant d(Λ)has a simple geometrical interpretation. In fact it is the
volume of the parallelotopeΠ, consisting of all pointsy∈Rnsuch that

y=θ 1 x 1 +···+θnxn,

where 0≤θk≤ 1 (k= 1 ,...,n). The interior ofΠis afundamental domainfor the
subgroupΛ,since