A note on Dedekind Criterion

A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1 is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Some well-known classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z and p-adic integers Zp and maximal orders of local fields and fields of algebraic numbers go back to classical papers by S. S. Ryshkov, P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the representations, and one of the possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n. The basic idea underlying a geometric approach was given in Ryshkov’s papers on the computation of the finite subgroups of GLn(Z) and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached. In the present paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of the group G. A combinatorial construction of the series of these representations is given explicitly. In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions is constructed. We consider certain related questions including the embedding problem in Galois theory for local faithful primitive representations of supersolvable groups and integral representations arising from elliptic curves.

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.

Computing GCDs of Polynomials over Algebraic Number Fields

In an algebraic number field K the ring Jk, generated by the k-th powers of all integers of K, and the additive group Ĵk, generated by the k-th powers of all primes of K, are investigated. Necessary and sufficient conditions for the case Ĵk=J1 are given. When k is prime to the discriminant of K a simple characterization of Jk is found and the case Ĵk=Jk is settled. Analogous results are obtained if one considers algebraic integers and primes of fixed signatures.

The purpose of this article to determine and classify the indecomposable RG-lattices, where R is a certain ring of algebraic integers, and G is a cyclic group of prime order.

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

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

Let K be an algebraic number field with non-trivial class group G and OK be its ring of integers. For k 2 N and some real x 1, let Fk(x) denote the number of non-zero principal ideals aOK with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk(x) behaves, for x ! 1, asymptotically like x(logx) 1/|G| 1 (loglogx) Nk(G) . In this article, it is proved that for every prime p, N1(Cp Cp) = 2p, and it is also proved that N1(Cmp Cmp) = 2mp if N1(Cm Cm) = 2m and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that N1(Cmn Cmn) = 2mn. Our results confirm a conjecture given by W. Narkiewicz thirty years ago partly and improve the known results substantially.

In many simple integral domains, such as Z or Z[i], there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like Z[x] or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we will show how to computably extend Z in such a way that we can control the ordinary integer primes in any Π20 way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable unique factorization domain (UFD) such that the set of primes is Π20-complete in every computable presentation.

Clean group rings over localizations of rings of integers

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a∈H and each two factorizations z,z ′ of a, there exist factorizations z=z 0 ,...,z k =z ′ of a such that, for each i∈[1,k], z i arises from z i-1 by replacing at most N atoms from z i-1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H)≤4.

In the previous chapter, we dealt with solvable groups and the methods came from abelian groups. In this chapter, we deal with \(S{L_2}({\mathcal{O}_S})\) where \(\mathcal{O}\) is the ring of integers in an algebraic number field and S is any set of places containing all the archimedean ones. The methods here will be very different from those of the previous chapter.

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) > 500.

Let θ be an algebraic integer over ℚ with minimum polynomial p(x) ∈ Z[x] of order n. Let θ = θ1,…, θn be the conjugates of θ. Assume that |θi| > 1 holds for i = 1,…, n. For an element a = α(θ) ∈ ℚ(θ) the conjugate with respect to θj will be denoted by α(θj). We shall use the notation p j = 1/θj (j = 1,…, n).