By J. W. S. Cassels (auth.)

Reihentext + Geometry of Numbers From the reports: "The paintings is punctiliously written. it really is good stimulated, and fascinating to learn, whether it's not consistently easy... ancient fabric is included... the writer has written a superb account of an enticing subject." (Mathematical Gazette) "A well-written, very thorough account ... one of the subject matters are lattices, relief, Minkowski's Theorem, distance services, packings, and automorphs; a few purposes to quantity conception; very good bibliographical references." (The American Mathematical Monthly)

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**Example text**

Then a*a = L ujvi is an integer. Now let c be any vector such that ca is an integer for all a in A. In particular (1~i~n) is an integer. Put a* = L ujbj. (c - a*) b j Then =0 (1~i~n); and so c =a* since the bi are linearly independent. This proves the first sentence of the theorem. The second sentence follows immediately from the first and also from (2). Finally. ) det(b1 . . . b and so d (A*) d (A) ll ) = 1. = 1. This concludes the proof of the lemma. 2. When Y=F0 is fixed. the points z such that yz=O lie in a hyperplane through o.

J are real numbers and the form h(x) = hZ2 x~ + 2h23 X2Xa + h33 xi must be positive definite. The determinant of h (x) is h22h33 - h~3 = - (1 _1])3 D = (1 -1])31 DI. After a transformation on the variables x 2 , x3 , we may suppose that h(x) is reduced; and so ( 10) by Theorem II. x z)2 + h22X~, of determinant - h22 • Clearly M(G) ~ (1 -1]) M(g) = 1 -1]. Hence, by Theorem IV, either h ~ 221 22- 100 (1 _ 1] )2 ( t 1) Indefinite quadratic forms 47 or G(Xl' x 2) is equivalent to one of t(Xi+XIX2-X~) or t(xl-2xi) for some number t with 1tl ~ (1-17).

B" is a basis for /\, then the general point b = U 1 b 1 + ... , ... (u1 b l + ... bl + ... b". bl , ... b", and d (a. A) = Idet (a. b l , ... , a. ) II det (bl , ... )1 d (A) . ° We note two particular cases. First, if t =1= is a real number, then the set of tb, bEA is a lattice of determinant Itlnd(A) which we shall denote by tl\. Ao, where a. is of the type (1) and Ao is the lattice of integer vectors. For if flt, ... ,a" is any basis for A, we may define IXi; by a; = (lXii' ... 4. Forms and lattices.