Explicit integral basis of [formula omitted

Let K=ℚ(𝜃) be an algebraic number field with 𝜃 satisfying an irreducible polynomial x6−m having integer coefficient m. We explicitly provide an integral basis of K along with some examples. As an application, we show that when m is a squarefree integer such that m≡2,3 mod4 and m≢±1 mod9, then K has a power integral basis.

On the modular forms of weight 1/2 over algebraic number fields

Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and N - α is a multiple of p - α for each prime divisor p of N. By the Korselt set of N, we mean the set of all α ∈ ℤ\{0} such that N is a Kα-number; this set will be denoted by [Formula: see text]. Given a squarefree composite number, it is not easy to provide its Korselt set and Korselt weight both theoretically and computationally. The simplest kind of squarefree composite number is the product of two distinct prime numbers. Even for this kind of numbers, the Korselt set is far from being characterized. Let p, q be two distinct prime numbers. This paper sheds some light on [Formula: see text].

In this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

Let Q denote the rational number field and α be an algebraic number of degree s. Then the algebraic number field F s = Q(α) is the field given by the polynomials in α of degree < s with rational coefficients.

Let be an algebraic number field with θ a root of an irreducible trinomial belonging to In this paper, for each prime number p we compute the highest power of p dividing the discriminant of K in terms of the prime powers dividing and discriminant of f(x). An explicit p-integral basis of K is also given for each prime p and a method is described to obtain an integral basis of K from these p-integral bases which is illustrated with examples.

Even-order subgroups of finite-dimensional division rings

We are interested in finding the relation between rings of endomorphisms of toroidal groups and algebraic number fields which are not totally real. For algebraic number fields K , we define real maximal algebraic number fields K 0 and 𝔬 K 0 -quasi-abelian varieties. We show that any simple 𝔬 K 0 -quasi-abelian variety is constructed by embeddings of a non-totally real algebraic number field K and it has finite dimensional cohomology groups.

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) &gt; 500.

Square-free values of cubic polynomials in algebraic number fields

In 1924 Hasse proved a fundamental theorem concerning quadratic forms over algebraic number fields, namely that two quadratic forms with coefficients in an algebraic number field K are equivalent if and only if they are so in every completion K p of K by valuations of K.

A characterization of number fields by p-closed extensions

Let p be a prime number where p ≡ 2 (mod 3). In this work, we give a nonnegative integer solution for the Diophantine equation 3x+py = z2. If y = 0, then (p, x, y, z) = (p, 1, 0, 2) is the only solution of the equation for each prime number p. If y is not divisible by 4, then the equation has a unique solution (p, x, y, z) = (2, 0, 3, 3). In case that y is a positive integer that is not divisible by 4, we give a necessary condition for an existence of a solution and give a computational result for p &lt; 1017. We also give a necessary condition for an existence of a solution for qx + py = z2 when p and q are distinct prime numbers.

Let <TEX>${\alpha}$</TEX> be a positive integer, and let <TEX>$p_1$</TEX>, <TEX>$p_2$</TEX> be two distinct prime numbers with <TEX>$p_1$</TEX> < <TEX>$p_2$</TEX>. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form <TEX>$2^{\alpha}p_1p_2$</TEX> and <TEX>$2^{\alpha}p_1^2p_2$</TEX>, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form <TEX>$p_1=2^{{\alpha}+1}-1$</TEX> and <TEX>$p_2={\frac{p^2_1+p_1+1}{3}}$</TEX>, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.

In this paper, we study a classical construction of lattices from number fields and obtain a series of new results about their minimum distance and other characteristics by introducing a new measure of algebraic numbers. In particular, we show that when the number fields have few complex embeddings, the minimum distances of these lattices can be computed exactly.