By C. G. Lekkerkerker, N. G. De Bruijn, J. De Groot, A. C. Zaanen

This quantity features a relatively whole photograph of the geometry of numbers, together with kin to different branches of arithmetic akin to analytic quantity idea, diophantine approximation, coding and numerical research. It offers with convex or non-convex our bodies and lattices in euclidean area, etc.

This moment version used to be ready together by means of P.M. Gruber and the writer of the 1st version. The authors have retained the present textual content (with minor corrections) whereas including to every bankruptcy supplementary sections at the more moderen advancements. whereas this system could have drawbacks, it has the yes benefit of displaying sincerely the place fresh growth has taken position and in what components fascinating effects can be anticipated within the future.

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For each λ, the body XK contains at most finitely many lattice points. Therefore, for ι; = 1 , . . , n, the closure λιΚ contains the same set of lattice points as the body XK, if λ > Àt and λ is sufficiently near to λι. Hence, λιΚ contains at least i" independent lattice points. Similarly, int λχΚ contains/— 1 ^ /— 1 independent lattice points, if y is the lowest index with λί = λ^ It follows from these remarks that we can choose successively a point t/1 in λχΚ, a point u2 in λ2Κ, . . 1"1) (i = 1 , .

1). Thus the left hand member of (15) is at least 1. So we have Theorem 5. } 2 . (16) It is not difficult to show that the right hand member of (16) is always greater than 1. This proves that the discriminant D is always greater than 1 in absolute value. Instead of AT we may use the domain K': m ^ y (17) tfr+J\ = | ^ + S + J | ύ rr+J where Tt,.. ,rr+s Ü - 1 , . ·, r) (j = 1 , . . , s), are arbitrarily given positive numbers. We have V(K') = 2'(2πγτιτ2 · ■ ■ τ,(τ, + 1 · · · τ Γ + 5 ) 2 ΜΓ χ . Consequently, K' contains a lattice point u Φ o, that is, F contains an algebraic integer ω Φ 0 with | ω 0 ) | ^ τ5 (j = 1, .

F(n) generated by 3(1\ 9 ( 2 ) , . . , $(n) respectively, are called the conjugate fields of F. If all # 0 ) are real, the number #, and also the field F, are called totally real. If all fields F(j) coincide, or, what comes to the same thing, if each 9U) can be expressed as a polynomial in 9 with rational coefficients, then F is called a normal field. Next, if ξ = P o + P i $ + ' ' ' +Pn-i$ r t ~ 1 is a n v number of F, the n numbers (3) ί"> = ρ0 + Ριθ (Λ + · · · +Pn-1(Z(JTl U = 1. · · ·, ») are called the field conjugates of {.