Let K be an abelian extension of the rationals. Let S(K) be the Schur group of K and let CC(K) be the subgroup of S(K) generated by classes containing cyclic cyclotomic algebras. We characterize when CC(K) has finite index in S(K) in terms of the relative position of K in the lattice of cyclotomic extensions of the rationals.

‎in 1970‎, ‎menegazzo [gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali‎, ‎ atti accad‎. ‎naz‎. ‎lincei rend‎. ‎cl‎. ‎sci‎. ‎fis‎. ‎mat‎. ‎natur. 48 (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $im$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎a group $g$ is said to have the $fm$...

We construct explicitly APF extensions of finite extensions of Qp for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with Zp-quotient. Let K be a finite extension of Qp and L/K be a Galois totally ramified pro-pextension. If the Galois group of L/K is a p-adic Lie group it is known from Sen ([Sen]) that the sequence of upper ramification breaks of L/K i...

Definition. An algebraic integer is the root of a monic polynomial in Z[x]. An algebraic number is the root of any non-zero polynomial in Z[x]. We are interested in studying the structure of the ring of algebraic integers in an algebraic number field. A number field is a finite extension of Q. We’ll assume that the number fields we consider are all subfields of C. Definition. Suppose that K and...

We develop the theory of “branch algebras”, which are infinitedimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field k we construct a k-algebra K which • is finitely generated and infinite-dimensional, but has only finite-dimensional qu...

We show that there is essentially a unique elliptic curve $E$ defined over cubic Galois extension $K$ of $\mathbb Q$ with $K$-rational point order 13 and such not Q$.

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. If K is not locally finite, the rank of A over K is infinite.

Let E/K be an elliptic curve defined over a number field, let ĥ be the canonical height on E, and let K/K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E/K) > 0 so that every nontorsion point P ∈ E(K) satisfies ĥ(P ) > C(E/K).

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. Unless K ⊂ F̄p for some p, the rank of A over K is infinite.

Let K be a field of characteristic 6= 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. Unless K ⊂ F̄p for some p, the rank of A over K is infinite.