Characters and Cyclotomic Fields in Finite Geometry

1.1 The nature of the problems 1.2 The combinatorial structures in question 1.2.1 Designs 1.2.2 Difference Sets 1.2.3 Projective planes and planar functions 1.2.4 Projective geometries and Singer difference sets 1.2.5 Hadamard matrices and weighing matrices 1.2.6 Irreducible cyclic codes, two-intersection sets and sub-difference sets 1.3 Group rings, characters, Fourier analysis 1.4 Number theoretic tools 1.5 Algebraic-combinatorial tools

The aim of this note is to study the class group of a central Ω-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Ω-Krull ring. In [8] we defined an Ω-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central Ω-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of Ω-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[(Xi)i∊I] (I may be infinite) is a central Ω-Krull ring.

We obtain an upper bound for the absolute value of cyclotomic integers which has strong implications on several combinatorial structures including (relative) difference sets, quasiregular projective planes, planar functions, and group invariant weighing matrices. Our results are of broader applicability than all previously known nonexistence theorems for these combinatorial objects. We will show that the exponent of an abelian group G containing a (v,k,ג,n)-difference set cannot exceed ((2^(s-1).F(v,n))/n)^0.5where is the number of odd prime divisors of v and F(v,n) is a number-theoretic parameter whose order of magnitude usually is the squarefree part of . One of the consequences is that for any finite set P of primes there is a constant C such that exp(G) ≤ C|G|^0.5for any abelian group G containing a Hadamard difference set whose order is a product of powers of primes in P. Furthermore, we are able to verify Ryser's conjecture for most parameter series of known difference sets. This includes a striking progress towards the circulant Hadamard matrix conjecture. A computer search shows that there is no Barker sequence of length l with 13< l <4x10^12. Finally, we obtain new necessary conditions for the existence of quasiregular projective planes and group invariant weighing matrices including asymptotic exponent bounds for cases which previously had been completely intractable.

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

Infinite families of cyclotomic function fields with any prescribed class group rank

A ring [Formula: see text] is said to be clean if each element of [Formula: see text] can be written as the sum of a unit and an idempotent. [Formula: see text] is said to be weakly clean if each element of [Formula: see text] is either a sum or a difference of a unit and an idempotent, and [Formula: see text] is said to be feebly clean if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text] are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] are distinct primes. In this paper, we consider a more general setting. Let [Formula: see text] be an algebraic number field, [Formula: see text] its ring of integers, [Formula: see text] a nonzero prime ideal and [Formula: see text] the localization of [Formula: see text] at [Formula: see text]. We investigate when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when [Formula: see text] is a cyclotomic field or [Formula: see text] is a quadratic field.

A new construction method for codes using encodings from group rings is described and presented. They consist primarily of two types: zero-divisor and unit derived codes. Previous codes from group rings focused on ideals; for example, cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. The authors use the result that a group ring is isomorphic to a certain well-defined ring of matrices and thus, every group ring element has an associated matrix. This allows matrix algebra to be used as needed in the study and production of codes, enabling the creation of standard generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties, such as being LDPC or self-dual, may be expressed as properties within the group ring, thus enabling the construction of codes with these properties. The methods are general enabling the construction of codes with many types of group rings. There is no restriction on the ring and thus codes over the integers, over matrix rings or even over group rings themselves are possible and fruitful.

In the previous papers1-2 we constructed an -dimensional Hopf algebra which is isomorphic to Drinfeld quantum double of Taft's Hopf algebra if and is a primitive nth root of unity, and studied the irreducible representations of . In this paper, we continue our study and examine the irreducible representations of when q is any nth root of 1. We give the structures of all simple -modules, and then classify them. We show that there are only distinct simple -modules up to isomorphism. Though is not a bialgebra, we show that the tensor product of two simple -modules possesses an -module structure in away similar to the case that q is a primitive nth root of 1. Then we give a sufficient and necessary condition for the tensor product of two simple -modules to be semisimple. More generally, we describe the structure of socal of the tensor product. Quantum groups arose from the study of quantum inverse scattering method, especially the Yang-Baxter equation. Quantum groups added new aspects to representation theory. Let be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. It was shown that R can be derived from a left H-module structure and a right H-comodule structure on M for some bialgebra H over k (see[3], [5]. The module and comodule structures satisfy a natural compatibility condition. M together with this structure is called a quantum Yang-Baxter H-module, which is also called a Yetter-Drinfeld module over H (see6-7. A quantum Yang-Baxter H-module is a (left) crossed H-bimodule.[5] Let H be a finite-dimensional Hopf algebra over afield k, and let be the Drinfeld's quantum double derived from H.[8] It is well known9-10 that a vector space M possesses a crossed H-bimodule structure if and only if M possesses a left -module structure. Hence the Yetter-Drinfeld module category is the same as the left -module category Suppose that and that is a primitive nth root of unity. Then is also a primitive nth root of unity. Taft constructed an -dimensional Hopf algebra in[11]. The 's form an interesting class of pointed Hopf algebras from a combinatorial point of view. When n is odd, provides an invariant of three-manifolds.[12] Generally, the double of is of interest in connection with knot theory. Kauffman and Radford showed[13] that is a ribbon Hopf algebra if and only if n is odd. In the paper,[1] the author constructed an infinite-dimensional noncommutative and noncocommutative Hopf algebra for any with . When q is a root of the nth cyclotomic polynomial over Z, has an -dimensional quotient Hopf algebra . If q is a primitive nth root of unity, then is isomorphic to as a Hopf algebra for any . If q is just a nth root of 1, then is merely an algebra.In[2], the author examined the irreducible representations of , equivalently, of , where and q is a primitive nth root of 1. In this paper, we will consider the more general case that q is any nth root of 1, and examine the irreducible representations of the algebra . In this case, let m be the order of q, then , and is an algebra which is not necessarily a bialgebra (Hopf algebra). Let . In Sec. 2, we give the classification of the simple -modules. We show that for any and there exists a unique l-dimensional simple -module , and that for any and , there exists a unique m-dimensional simple -module . We also show that any simple -module is either isomorphic to some or isomorphic to some . Upto isomorphisms, there are only distinct simple -modules. Though is not a bialgebra, we will show that the tensor product of two simple -modules possesses an -module structures which is similar to the case[2] that is a Hopf algebra. In Sec. 3, we consider the tensor product of two simple -modules U and V, and give a sufficient and necessary condition for the tensor product to be semisimple. We first show that if U and V are simple -module with dim or dim then is also a simple -module. Then we prove that is a semisimple -module if and only if , that with is semisimple if and only if , and that is semisimple if and only if or (mod m). Generally, we describe the structure of socal of the tensor product. In particular, if we take , then we can get all the results described in[2].

Termatiko sets are combinatorial structures that have been shown to hinder the success of the Interval-Passing Algorithm in compressed sensing. In this paper, we show how termatiko sets relate to other combinatorial structures in graphs representing measurement matrices that are also known to cause failure in similar iterative algorithms. We give bounds on the sizes of termatiko sets of measurement matrices based on finite geometries and also investigate the effect of the redundancy of the matrices on the number of these sets.

Let Cq denote the cyclic group of order q and ZCq the integral group ring of Cq. If q is a prime, q = p say, D. S. Rim [18] has proved that the projective class group ITio(ZCp) is isomorphic to /(o(Z[~]), where ~ denotes a primitive p-th root of unity. In turn, it is well known that/(o(Z[~]) is isomorphic to the ideal class group of the ring Z[~] of integers in the cyclotomic field Fo = Q(~). See J. Milnor's book [17], w Corollary 1.11. In this paper we study fflo(ZCq) for q = p,+l, where p is a prime number. For instance, we obtain in w the following result. Let C(Fn) denote the ideal class group of the cyclotomic field F, = Q(~,), .where ~, is a primitive pn+l-st root of unity. If p is a semi-regular odd prime, there is an exact sequence

Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information about $G$ is encoded in all the twisted group rings of $G$ over $R$. We investigate this problem over fields. We start with abelian groups and show how the results depend on the roots of unity in $R$. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when $R$ is not an algebraically closed field, but still linearizes all projective representations of a group. We then show that groups from the celebrated example of Everett Dade which have isomorphic group algebras over any field can be distinguished by their twisted group algebras over finite fields.

On the maximal quotient ring of regular group rings

For a field, a normal extension of is a field F containing such that the group of automorphisms of F leaving point-wise fixed (the Galois group) is finite and leaves no more than fixed. It is an untouched classical problem to determine the normal extensions of k. Because of this, realistic work has centered on finding the Abelian extensions (the normal extensions where the Galois group is commutative). Where is an algebraic number field this makes up the class field theory. Where is any field of characteristic zero containing all the roots of unity, the Abelian extensions are given by the Kummer theory. In this paper we generalize the Kummer theory to an arbitrary field. In the characteristic p case or in the case where roots of unity do not exist, our answer, although it does not involve field extensions and thus is technically correct, is not as explicit as we could wish. For instance, it is not clear how to use this work in the derivation of the class field theory. Yet our answer is in terms of a cohomology theory in which a great deal of machinery exists simply because the theory is exactly analogous to (i.e., is the same in category theory as) the cohomology of groups, and for this reason we feel it presents a natural, systematic approach to questions involving Abelian extensions in the same way that the less general Kummer theory provides such an approach. If we replace the word set by commutative with identity over k and the phrase map from A to by algebra homomorphism from B to A, then the concept of a group transforms (using category theory for precision) to an object which is often called a group scheme (or equivalently, a Hopf with inverse map). The ordinary cohomology of groups, together with all its formalistic properties, transforms to these schemes. The group rings k(H) and k(L) are such schemes, where H denotes the group of integers and L denotes the rationals modulo the integers. Our result is that the second cohomology group of k(L) with coefficients in k(H) (trivial operation) is naturally isomorphic to the character group of the full Galois group of k. This means that the Abelian extensions of are in one-one correspondence with the finite subgroups of of H2(k(L), k(H)). This cohomology group is explicitly a certain factor group of

The aim of this thesis is to investigate the circumstances under which group rings over fields have non-zero socle, i.e. contain minimal one-sided ideals. After an introductory chapter, we consider the special case o₊ a periodic abelian group and a non-modular field (that is, a field of characteristic prime to the orders of the elements of the group). This special case, and the background material contained in Chapter III, serve as preparation for our principal results, which concern, locally finite groups. We establish necessary and sufficient conditions on an arbitrary field K and a locally finite group 0 for the group ring KG to contain minimal one-sided ideals: the most important condition is that G should be a Cernikov group. We then examine the structure of KG when these conditions are satisfied. Vie show that KG has a finite series of ideal3 each factor of which i3 a direct sum of quasi-Frobenius rings, and characterize the socle of KG. Me also classify indecomposable KG-modules, and determine (for countable but not necessarily locally finite groups G) necessary and sufficient conditions for all indecomposable KG-modules to be irreducible. In the final chapter we consider non-locally-finite groups, conjecturing that group rings of such groups never contain minimal one-sided ideals. We establish the truth of this conjecture for several classes of groups, and also consider semiartinian group rings.

We give a small, but very useful modification of a criterion of Mignotte ([4]) for Catalan’s equation, replacing the class number of a certain abelian field by the relative class number, which is much easier to compute. The proof is the same, apart from the idea to consider the class group modulo the ideals coming from the real subfield. We use the following notation: K is a CM-field, IK its group of fractional ideals and i : K∗ → IK the canonical map x 7→ (x); j denotes complex conjugation, K+ the maximal real subfield and h−(K) the relative class number of K; OK is the ring of integral elements of K. Lemma 1. Let K be a CM-field and Q a finite set of prime ideals of K. There is a subgroup I0 of the ideal group IK such that (i) the prime ideals in Q do not appear in the factorization of any ideal in I0; (ii) IK/(i(K∗)I0) has cardinality h−(K) or 2h−(K); (iii) if e ∈ K∗ with (e) ∈ I0, then e1−j is a root of unity. P r o o f. Let I0 consist of those ideals which are in the image of the canonical map IK+ → IK , and which do not contain any prime ideal in Q. If (e) ∈ I0, then (e) = (e), so e1−j is a unit, hence also a root of unity because all its conjugates have absolute value 1 (cf. [6], Lemma 1.6). It remains to show (ii). It is an easy consequence of the approximation theorem that every ideal class contains an ideal without primes in Q (see e.g. [3], IV, Corollary 1.4). Therefore IK/(i(K∗)I0) = ideal class group of K modulo image of the ideal class group of K+. By [6], Theorem 10.3, at most 2 ideal classes of K+ become principal in K, so the statement follows. Theorem 1. Let p 6= q be odd prime numbers. Let ζ be a primitive p-th root of unity and K an imaginary subfield of L := Q(ζ). Catalan’s equation x − y = 1 has no nontrivial integral solution if q -h−(K) and pq−1 6≡ 1 mod q2.