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Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The present paper gives explicit construction of an integral basis of $K$ along with applications. This construction of an integral basis of $K$ extends a result proved in [J. Number Theory, {173} (2017), 129-146] regarding periodicity of integral bases of $\mathbb{Q}(\sqrt[n]{a})$ when $a$ is squarefree.

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequalityhas only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion., Comment: arXiv admin note: text overlap with arXiv:1203.1030

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure Kof K. For an element α e K\K, a chain α = α 0 , α 1 ,…,α r of elements of K,such that α i is of minimum degree over K with the property that ῡ(α i-1 -α i )= sup{ῡ(α i-1 -β)|[K(β): K] i -1):K]} and that α r e K , is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of K(cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure λ of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to λ is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

Let ν be a valuation of any rank of a field K with value group G ν and f(X)= X m + a l X m−1 + … + a m be a polynomial over K. In this paper, it is shown that if (ν(a i )/i)≥(ν(a m )/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(a m )/r∈G ν , then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

The problem concerning the distribution of the fractional parts of the sequence an k ( k an integer exceeding one) was first considered by Hardy and Littlewood [6] and Weyl [20] earlier this century. This work was developed, with the focus on small fractional parts of the sequence, by Vinogradov [17], Heilbronn [13] and Danicic [2] (see [1]). Recently Heath-Brown [12] has improved the unlocalized versions of these results for k ≥ 6 (a slightly stronger result than Heath-Brown's for K = 8 is given on page 24 of [8]. The method mentioned there can, after some numerical calculation, improve Heath-Brown's result for 8 ≤ k ≤ 20, but still stronger results have recently been obtained by Dr. T. D. Wooley). The cognate problem regarding the sequence ap k , where p denotes a prime, has also received some attention. In this situation even the case k = 1 proves to be difficult (see [9] and [14]). The first results in this field were given by Vinogradov (see Chapter 11 of [19] for the case k = 1, [18] for k ≥ 2). For k = 2 the best result to date has been supplied by Ghosh [5], and for ≥, by Harman (Theorem 3 in [9], building on the work in [7] and [8]). In this paper we shall improve the known results for 2 ≤ k ≤ 12. For larger k , Theorem 3 in [8] is more efficient. The theorem we prove is as follows.

Let V 0 be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of V 0 .In this paper, we determine explicitly all rank 2 valuations of K(x) which extend V 0 .One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.

In this paper we are interested in two related measures of the degree of approximation of a complex number ζ by algebraic numbers. For a given integer n ≥ 1, write wn(ζ) for the supremum of the exponents w for whichfor infinitely many polynomialsin Z[x] of height H(P) = max |av|. Clearly 0 ≤ w1 (ζ) ≤ w2 (ζ) ≤ …. On the other hand, write for the supremum of the exponents w for whichfor infinitely many algebraic numbers α of degree at most n.

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.

Let Κ be a finite number field and let ο be the ring of algebraic integers in Κ. The algebraic integers in a finite extension field Λ of Κ form a ring . We shall be concerned here with the structure of such rings , viewed as modules over ο. It will be useful to begin with a brief discussion of a new concept of the discriminant of Λ/Κ, introduced in a preceding paper [1], which will be our principal tool (see also [2]).

In a recent paper [4], I introduced the notion of recursive formal Lie groups (of infinite dimension) over a field of characteristic p > 0, and studied a particular class of such groups, the groups of hyperexponential type ; these can be characterized either as being (recursively) isomorphic to a special group of that class, the hyperexponential group , or by simple conditions on their Lie hyperalgebra. An interesting example of a group of that class is the additive Witt group W , whose “infinitesimal” structure can therefore be considered as known, at least “up to an isomorphism”. However, the intrinsic importance of the Witt group (which, as well known, is the “formalization”, so to speak, of the additive group of a p-adic field) leads one to think that it may be worth while to study in greater detail that group itself, instead of being content with the mere existence of an unspecified isomorphism with the hyperexponential group. This is what we intend to do here; it turns out that, although it seems hopeless to write down explicitly the group law of the Witt group, the multiplication table of its hyperalgebra is, on the contrary, as simple as one could hope, and is, in fact, identical to that of the hyperalgebra of the hyperexponential group (although the two groups are distinct). Moreover, this leads to a new and quite unexpected definition of the Witt group, which links it still closer to the hyperexponential group, and provides a well-determined isomorphism between the two groups.

The object of this paper is first to generalize the basic inequality of the large sieve method to exponential sums in many variables, and then to deduce results for algebraic number fields that are analogous to known results for the rational field.