Simultaneous rational approximations to certain algebraic numbers

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

Linear forms at a basis of an algebraic number field

The only easy result here is as follows. Suppose†) 1,α 1,...,α v are algebraic and linearly independent over the rationals. Let d be the degree of the number field generated by α 1,...,α v and let 1,α 1,...,α v,...,α d−1 be a basis of this field. We saw in Theorem 4A of Chapter II that α 1,...,α d−1 are badly approximable, so that $$ \left| {\alpha _1 q_1 + \cdots + \alpha _{d - 1} q_{d - 1} - p} \right| > c_1 q^{ - d + 1} $$ where q1,...,qd−1, p are rational integers and where q = max(|q1|,...,|qd−1|) ≠ 0. Taking qv+1 = ... = qd−1 = 0, we have LEMMA 1A. Suppose 1,α 1,...,α v are linearly independent over ℚ, and they generate an algebraic number field of degree d. Then $$ \left| {\alpha _1 q_1 + \cdots + \alpha _v q_v - p} \right| > c_1 q^{ - d + 1} $$ for arbitrary integers q1,...,qv, p having q = max(|q1|,..., |qv|) > 0.

As to the meaning of the symbols, the reader is referred to the introduction of Part I. We recall here only that F is a totally real algebraic number field of degree n, and w denotes the Fourier coefficients of an elliptic modular form 52(z) = E wQ(a)e2q'iaz. In Part I, we investigated D assuming that EX(a)N(Q) S is an L-function of a CM-field with an algebraic-valued Hecke character. In the present Part II, we treat the case where X( a) can be obtained as eigenvalues of Hecke operators T(a) on a quaternion algebra B over F. Thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane & with respect to congruence subgroups of B x . Here r is the number of archimedean primes of F unramified in B. Then we introduce in Section 2 the notion of arithmeticity (or Q-rationality) of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones. Our main theorems will be stated in Section 3 and proved in Section 6; Sections 4 and 5 are preliminaries to the proof. The central result, specialized to the case r = 1, asserts that if g is a Q-rational eigenform such that gj T( a) =X( a )g for all integral ideals a of F, then D(s0) for certain integers so are algebraic numbers times 7Tdg, g), where d is an integer determined by the weight of g, and (*, *) is the Petersson inner product.

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

Let K be an algebraic number field of degree n + 1 over the rational field R. The conjugates of K are denoted by K(?), , I K(n), it being supposed that K M K(cl are real, K(r?l+), * , K(n) are complex, and moreover that for k =r1+ *, ri +r2 the fields K(k) and K(k+r2) are obtainable from one another by interchanging i( = (1)1/2) and -i (here r1+2r2=n). I assume throughout that r1 ?0 and that n ?1. Numbers in K are represented by small Greek letters, the insertion of superscripts denoting the passage to the corresponding conjugates. Let co, * * , wn be a set of numbers in K which are linearly independent over R; let co = 1; and finally let f(x1, * * *, xn) be a (complex-valued) function of n real variables which is Riemann integrable over the unit cube En: O<Xk<l (k=1, * * *, n). Then, according to theorems of Weyl on uniform distribution,

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

The purpose of this paper is to prove the following result.Theorem 1. Let K be a real algebraic number field of degree m = n + 1. Let 1, β1, …, βn be a basis of K.

It is not known whether or not the Jacobi-Perron Algorithm of a vector in ${R_{n - 1}},n \geqslant 3$, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied,

The Minkowski method of unit search is applied to particular types of parallelotopes permitting to discover algebraic integers of bounded norm in a given algebraic number field of degree n at will by solving successively 2n linear inequalities for one unknown each.Application is made to the unit search for all totally real number fields of minimal discriminant for n < 7.Introduction.Many methods have been devised for producing maximal sets of independent units of an integral domain 0 with a finite basis cj, , . . ., con over the rational integer ring, Z.The number geometric methods devised so far always lead to a large number of linear inequalities for n unknowns simultaneously.It is the purpose of this note to reduce the application of number geometric methods to sequences of n pairs of linear inequalities, each only for one unknown.1. Parallelotopes of Bounded Norm.Denoting by r, the number of distinct isomorphisms of 0 into the real number field, R, say

A note on class numbers of algebraic number fields

It is our purpose here to show that, using results already in the literature, it is easy to prove the following and similar theorems. For every positive integer d, there exists an integer Ψ ( d ) such that if K is an algebraic number field of degree d over the field of rational numbers then every cubic form f(x 1 x 2 , …, x n ) over K, with n ≥ Ψ(d), has a non-trivial zero in K.

The Minkowski method of unit search is applied to particular types of parallelotopes permitting to discover algebraic integers of bounded norm in a given algebraic number field of degree n at will by solving successively 2 n 2n linear inequalities for one unknown each. Application is made to the unit search for all totally real number fields of minimal discriminant for n ⩽ 7 n \leqslant 7 .

We show for the first time how to implement cryptographic protocols based on class groups of algebraic number fields of degree > 2. We describe how the involved objects can be represented and how the arithmetic in class groups can be realized efficiently. To speed up the arithmetic we present our new method for multiplication of ideals. Furthermore we show how to generate cryptographically suitable algebraic number fields. Besides, we give a numerical example and analyse our run times.

The structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.