On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

Class numbers of definite binary quadratic lattices over algebraic function fields

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 $\text{K}\text{k}$, 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 $\text{g(}\text{K}\text{k}\text{)}$ (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.     

A note on l-class groups of certain algebraic number fields

The structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number fields whose Galois groups are of type (p, p); (ii) non-Galois extensions $\text{Q(p}\text{d}{}_0\text{3}\text{,p}\text{d}{}_1\text{3}\text{)}$ of degree p² over Q; (iii) dihedral extensions of degree 2^{n + 1} over Q. It is shown that it is possible to obtain class number relations by group-theoretic methods. Subgroups of ideal class groups whose orders are prime to the extension degree are considered.     

Corresponding residue systems in normal extensions

Let K and K′ be number fields with L = K · K′ and F = KφK′. Suppose that $\text{K}\text{F}$ and $\text{K′}\text{F}$ are normal extensions of degree n. Let $\text{B}$ be a prime ideal in L and suppose that $\text{B}$ is totally ramified in $\text{K}\text{F}$ and in $\text{K′}\text{F}$. Let π be a prime element for $\text{B}$_K = $\text{B}$ φ K, and let f(x) ∈ F[x] be the minimum polynomial for π over F. Suppose that $\text{B}$_K · $\text{D}$_L = ($\text{B}$^≠)^e. Then,

$\text{M(B\# : K, K′) =}\text{min}\text{{m, e(t + 1)}}$

, where $\text{t =}\text{min}\text{{t(}\text{K}\text{F}\text{), t(}\text{K′}\text{F}\text{)}}$ and m is the largest integer such that ($\text{B}$_K′)^nm/e φ f($\text{D}$_K′) ≠ {φ}.If we assume in addition to the above hypotheses that [K : F] = [K′: F] = pⁿ, a prime power, and that $\text{B}$ divides p and is totally ramified in $\text{L}\text{F}$, then

$\text{M(B\# : K, K′) ? p}{}^{\mathrm{n?1}}\text{[(p ? 1)(t + p]}$

, with t = t($\text{B}$ : L/F).     

Rational compositum genus for a pure cubic field

In non-Abelian fields, to which genus theory does not ordinarily apply, many of the diophantine by-products are still available. For instance, the representability of primes as norms of principal ideals in different fields of the same degree will bear interrelations if the fields belong to a small compositum as do the four subfields of $\text{Q}$(m_∞^1/3, m₀^1/3). This is a generalization of a classical phenomenon for quadratic fields. It is particularly effective when just one prime l ≡ 1 (mod 3) divides m₀m_∞.     

Automorphisms and holomorphic differentials in characteristic p

Let G be a group of automorphisms of a function field F of genus g_F over an algebraically closed field K. The space $\text{Ω}{}_{\mathrm{F}}$ of holomorphic differentials of F is a g_F^? dimensional K-space. In response to a query of Hecke, Chevalley and Weil (Abh. Math. Sem. Univ. Hamburg, 10 (1934), 358–361) completely determined the structure of $\text{Ω}{}_{\mathrm{F}}$ as representation space for G in the classical case. They carried out the proof for the special case in which F is unramified over the fixed field of G. The case of cyclic ramified extensions had been previously considered by Hurwitz (Math. Ann., 41 (1893), 37–45). Weil (Abh. Math. Sem. Univ. Hamburg, 11 (1935), 110–115) gave a proof in the general case. The treatment in the last two papers is analytical. In characteristic p, the problem is open. If G is cyclic and F is unramified over the fixed field E of G, Tamagawa (Proc. Japan Acad., 27 (1951), 548–551) proved that the representation is the direct sum of one identity representation of degree 1 and g_E ? 1 regular representations. The principal object of this paper is an extension of Tamagawa's result to arbitrary cyclic extensions of p-power degree. The number of times an indecomposable representation of given degree occurs in the representation of G on $\text{Ω}{}_{\mathrm{F}}$ is explicitly determined in terms of g_E and the Witt vector characterizing the extension $\text{F}\text{E}$. The paper also contains a purely algebraic proof of the result of Chevalley and Weil for arbitrary cyclic extensions of degree relatively prime to p. Using character theory, it can be extended to arbitrary groups of order relatively prime to the characteristic.     

Permutation polynomials over algebraic number fields

Let K be an algebraic number field. It is known that any polynomial which induces a permutation on infinitely many residue class fields of K is a composition of cyclic and Chebyshev polynomials. This paper deals with the problem of deciding, for a given K, which compositions of cyclic or Chebyshev polynomials have this property. The problem is reduced to the case where K is an Abelian extension of $\text{Q}$. Then the question is settled for polynomials of prime degree, and the Abelian case for composite degree polynomials is considered. Finally, various special cases are dealt with.     

Class numbers of algebraic number fields of Eisenstein type,II

Let K be an algebraic number field, of degree n, with a completely ramifying prime p, and let t be a common divisor of n and $\text{(p ? 1)}\text{2}$. Then it is proved that if K does not contain the unique subfield, of degree t, of the p-th cyclotomic number field, then we have (h_K, n) > 1, where h_K is the class number of K. As applications, we give several results on h_K of such algebraic number fields K.     

On the rational equivalence of full decomposable forms

Let k be any finite normal extension of the rational field Q and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. Consider the set F of the full decomposable forms which correspond to the invertible fractional ideals of D. In a recent paper the author has given arithmetic criteria to determine which classes of improperly equivalent forms in F integrally represent a given positive rational integer m. These criteria are formulated in terms of certain integer sequences which satisfy a linear recursion and need only be considered modulo the primes dividing m. Here, for the most part, we consider partitioning F under rational equivalence. It is a found that the set of rationally equivalent classes in F is a group under composition of forms analogous to Gauss' and Dirichlet's classical results for binary quadratic forms. This leads us to given criteria as before to determine which classes of rationally equivalent forms in F rationally represent m. Moreover, by applying the genus theory of number fields, we find arithmetic criteria to determine when everywhere local norms are global norms if the Hasse norm principle fails to hold in $\text{k}\text{Q}$.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

The lattice of subfields of a radical extension

Let x^m ? a be irreducible over F with char $\text{F?m}$ and let α be a root of x^m ? a. The purpose of this paper is to study the lattice of subfields of $\text{F(α)}\text{F}$ and to this end $\text{C(}\text{F(α)}\text{F}\text{, k)}$ is defined to be the number of subfields of F(α) of degree k over $\text{F. C(}\text{F(α)}\text{F}\text{, p}{}^{\mathrm{n}}\text{)}$ is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then $\text{C(}\text{F(α)}\text{F}\text{, k) = C(}\text{F(α)}\text{F}\text{, (k, n)) = C(}\text{N}\text{F}\text{, (k, n))}$. The irreducible binomials x^s ? b, x^s ? c are said to be equivalent if there exist roots β^s = b, γ^s = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x^2e ? a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.     

Quaternionic compositum genus

A quaternionic field over the rationals contains three quadratic subfields with a compositum genus relation of the type described in the author's paper in Volume 9 of this journal, involving the representation of a prime as norm in these subfields. These representations had previously been only partially exlored by the transfer of class structure from the rational to the quadratic fields. Here a full exposition is given by constructing the Artin characters when the subfields are $\text{Q}$(2^1/2), $\text{Q}$(q^1/2), and $\text{Q}$(2q)^1/2 (q prime). A special role belongs to q = A² + 32b².     

Diophantine problems in variables restricted to the values 0 and 1

Let $\text{F}$x₁,…,x_s be a form of degree d with integer coefficients. How large must s be to ensure that the congruence $\text{F}$(x₁,…,x_s) ≡ 0 (mod m) has a nontrivial solution in integers 0 or 1? More generally, if $\text{F}$ has coefficients in a finite additive group G, how large must s be in order that the equation $\text{F}$(x₁,…,x_s) = 0 has a solution of this type? We deal with these questions as well as related problems in the group of integers modulo 1 and in the group of reals.     

Convex polyhedra of doubly stochastic matrices: II. Graph of Ωn

Properties of the graph $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ of the polytope $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices are studied. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is not a k-dimensional rectangular parallelotope for k ≥ 2, then G($\text{F}$) is Hamilton connected. Prime factor decompositions of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ relative to Cartesian product are investigated. In particular, if $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then the number of prime graphs in any prime factor decomposition of G($\text{F}$) equals the number of connected components of the neighborhood of any vertex of G($\text{F}$). Distance properties of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ are obtained. Faces $\text{F}$ of $\text{Ω}{}_{\mathrm{n}}$ for which G($\text{F}$) is a clique of $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ are investigated.     

On subgraph number independence in trees

For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If $\text{F}$ and $\text{G}$ are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F∈$\text{F}$, are linearly independent when G is restricted to $\text{G}$. For example, if $\text{F}$ = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and $\text{F}$ is the family of all (finite) trees, then of course N_K1(T) ? N_K2(T) = 1 for all T∈$\text{F}$. Slightly less trivially, if $\text{F}$ = {S_n: n = 1, 2, 3,…} (where S_n denotes the star on n edges) and $\text{G}$ again is the family of all trees, then Σ_n=1^∞(?1)ⁿ⁺¹N_Sn(T)=1 for all T∈$\text{G}$. It is proved that such a linear dependence can never occur if $\text{F}$ is finite, no F∈$\text{F}$ has an isolated point, and $\text{G}$ contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).     

On the class numbers of cyclotomic fields

Let g_n denote the first factor of the class number of the nth cyclotomic field. It is proved that if n runs through a sequence of prime powers p^r tending to infinity, then

$\text{log}\text{g}{}_{\mathrm{n}}\text{～}\text{1}\text{4}\text{[1 ? (}\text{1}\text{p}\text{)]n}\text{log}\text{n}$

.     

Patterns of power residues

This paper examines the question of whether a given pattern

$\text{x,x+a}{}_1\text{,…,x+a}{}_{\mathrm{m?1}}$

of kth power residues of length m can be postponed indefinitely. This is the case when there exists a prime q, called a delay prime, which does not contain this pattern even if q itself is considered as a kth power residue. It is conjectured that if there exists no delay prime then there exists a finite limit

$\text{Λ=Λ (k,m;a}{}_1\text{,…,a}{}_{\mathrm{m?1}}$

for which the corresponding pattern will occur before Λ in every sufficiently large prime of the form kn + 1.     

Corresponding residue systems in nonnormal extensions of prime degree

Let K₁ and K₂ be number fields and $\text{F = K}{}_1\text{? K}{}_2$. Suppose $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$ are of prime degree p but are not necessarily normal. Let N₁ and N₂ be the normal closures of K₁ and K₂ over F, respectively, L = K₁K₂, N = N₁N₂, and $\text{B}$ be a prime divisor of N which divides p and is totally ramified in $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$. Let ${}_{\mathrm{<mtext>N</mtext><mtext>L</mtext>}}$ be the ramification index of $\text{B}$ in $\text{N}\text{L}$, $\text{t}{}_{\mathrm{<mtext>L</mtext><mtext>F</mtext>}}$ be the total ramification number of $\text{B}$ in $\text{L}\text{F}$, and . Then $\text{M}$(K₁, K₂) is exactly divisible by $\text{B}$^M, where .     

Abelian extensions of topological vector groups

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group $\text{G}\text{F}$ are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and $\text{G}\text{F}$ a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if $\text{G}\text{F}$ is finite dimensional or if F is one dimensional and $\text{G}\text{F}$ a separable Hilbert group.