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Algebraic independence of values of certain infinite products is proved, where the transcendence of such numbers was already established by Tachiya. As applications explicit examples of algebraically independent numbers are also given.

Let AK denote the ring of algebraic integers of an algebraic number field K=Q(θ) where the algebraic integer θ has minimal polynomial F(x)=xn+axm+b over the field Q of rational numbers with n=mt+u, t∈N, 0≤u≤m−1. In this paper, we characterize those primes which divide the discriminant of F(x) but do not divide [AK:Z[θ]] when u=0 or u divides m; such primes p are important for explicitly determining the decomposition of pAK into a product of prime ideals of AK in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, n for AK to be equal to Z[θ].

In this paper, we iterate the algebraic etale-Brauer set for any nice variety X over a number field k with π 1 et ( X ‾ ) finite and we show that the iterated set coincides with the original algebraic etale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thelene on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k; moreover, it gives a partial answer to an “algebraic” analogue of a question by Poonen about iterating the descent set.

Let E/k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group Sn on n⩽5 letters. For any field extensions k⊆K, L⊆E a necessary and a sufficient condition is given for the equality N K/k K ∗ =N L/k L ∗ to hold, where N K/k K ∗ is the group of norms from K to k of the elements of the multiplicative group K ∗ of K.

We show that Diophantine problem (otherwise known as Hilbert's Tenth Problem) is undecidable over the fields of algebraic functions over the finite fields of constants of characteristic greater than two. This is the first example of Diophantine undecidability over any algebraic field. We also show that the Diophantine class of a holomorphy ring of an above mentioned algebraic function field does not change if the set of primes at which the functions of the ring are allowed to have poles is changed by adding or removing of finitely many primes.

In this paper a theory is developed which enables one to determine whether or not two algebraic numbers or two binary forms are equivalent over Z [ i ]. This extends earlier work of the author from the real case.

A generalization of the following theorem will be proved. If α is an algebraic number of degree n over Q and the trace of αi lies in Z for all positive i up to n + n log2n, then α is an algebraic integer.

Let K be an algebraic number field of finite degree over the rationals. The two themes of this paper are the problems about the existence of infinitely many power-free algebraic integers in polynomial sequences generated by irreducible polynomials of K with integer arguments and with prime arguments, respectively. In particular, it follows from the results that an irreducible cubic polynomial of K represents square-free numbers infinitely often, provided that an obvious necessary condition is fulfilled.

We extend some results of Christol and Furstenberg to the case of several variables: (1) A power series in several variables over the p-adic integers Z p is congruent mod ps to an algebraic power series if and only if its coefficients satisfy certain congruences mod ps. (2) Any algebraic power series in m variables over a field K can be written as the diagonal of a rational power series in 2m variables. We also give an elementary proof of a result of Deligne: The diagonal of an algebraic power series in several variables, over a field of nonzero characteristic is algebraic. Moreover we show that the diagonal of an algebraic power series over Z p is algebraic mod ps, for every s. We also obtain other related results.

The estimate from below of the modulus of the difference between π and algebraic numbers from the fields generated by the roots of unity is made.

The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds. Theorem . Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p. Theorem . Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b1 modulo p.

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions K k , class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus g( K k ) (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.

Let K be an algebraic number field with proper subfield k . If K and k have the same number of fundamental units then relations between the units of K and k are obtained.

Let Γ be an algebraic curve which is given by an equation f(x, y) = 0, f(x, y) ∈ k[x, y] where k is an algebraic number field and f(x, y) is irreducible. Suppose that there exists an ∗Γ a nonstandard point (ξ, η) ∈ ∗k × ∗k. Then k(ξ, η) is (isomorphic to) the algebraic function field of Γ and, at the same time, is a subfield of ∗k. Correlating the divisors of the function field k(ξ, η) and of the number field ∗k, we develop an analogue of the Artin-Whaples theory of the product formula. This leads to one of Siegel's basic inequalities for rational points on algebraic curves.

This paper discusses an application of Minkowski's theory of the successive minima in the geometry of numbers to the problem of the approximation of an algebraic or transcendental number a by algebraic numbers. I consider for simplicity only real numbers a . However, it is obvious that an analogous theory can be established for complex numbers, and also for p -adic numbers, as well as for the field of formal ascending or descending Laurent series with coefficients in an arbitrary field.

A finite extension K of an algebraic number field k is called k-solitary if for any finite extension L of k the equality of norm groups N K/k K* = N L/k L* implies that K and L are conjugate over k (i.e., K and L are k-isomorphic). In the present work we investigate finite Galois extensions of k which are k-solitary. We prove that if a finite Galois extension K of k is k-solitary, then K/k is a 2-extension. k and every quadratic extension of k is k-solitary. We establish a necessary condition for a finite Galois 2-extension K/k with (K : k) ≥ 4 to be k-solitary. We then investigate covering subgroups, and show that the necessary condition is also a sufficient condition for extensions of degree 4 or 8. We conclude our discussion by constructing a Galois extension E of the field of rational numbers Q with the Galois group isomorphic to the dihedral group of order 8 so that E is Q-solitary.

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thelene conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map H 1 (F,G)→∏ v∈Ω k H 1 (F v ,G) has trivial kernel, Ω k denoting the set of places of k. The conjecture is true if G is of type 1 A ∗ , i.e., isomorphic to SL1(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thelene, being an immediate consequence of the theorems of Merkurjev–Suslin [S1] and Kato [K] . Gille [G] proves the conjecture if G is defined over k and F=k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types 2 A ∗ , Bn, Cn, Dn (D4 nontrialitarian), G2 or F4; a group is said to be of type 2 A ∗ , if it is isomorphic to SU(B,τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.

We study the existence of non-special divisors of degree g and g - 1 for algebraic function fields of genus g ⩾ 1 defined over a finite field F q . In particular, we prove that there always exists an effective non-special divisor of degree g ⩾ 2 if q ⩾ 3 and that there always exists a non-special divisor of degree g - 1 ⩾ 1 if q ⩾ 4 . We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension F q n of F q , when q = 2 r ⩾ 16 .

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω1,…, ωn is an integral basis for the ring DL of integers in K, then the quadratic form Q whose matrix is (traceK∥F(ωiωj)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.

The stufe, s = s(K), of a field K is the least number such that −1 is the sum of s squares of elements of K; then every element of K is the sum of s + 1 squares. Using the Hasse-Minkowski theorem on quadratic forms, and a simple algebraic identity, one can easily show that the stufe of an algebraic number field, if it exists, is 1, 2, or 4. Here, a simple criterion is presented for determining s(K) for any algebraic number field K and is applied to the quadratic and cyclotomic cases. Clearly, s(K) exists only when K is totally complex, and equals 1 if and only if (−1) 1 2 ∈ K .

Let K be a number field of degree n with ring of integers OK. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if h∈K[X] maps every element of OK of degree n to an algebraic integer, then h(X) is integral-valued over OK, that is, h(OK)⊂OK. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial f∈Q[X]: if f(α) is integral over Z for every algebraic integer α of degree n, then f(β) is integral over Z for every algebraic integer β of degree smaller than n. This second result is established by proving that the integral closure of the ring of polynomials in Q[X] which are integer-valued over the set of matrices Mn(Z) is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to n.

It is shown that there are algebraic integers, with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Davenport for algebraic numbers.