Related papers: Relative Galois module structure of rings of integ…

We prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give…

We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property that $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-module. The work of Greither, Replogle, Rubin, and…

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O_L is free as an O_K[G]-module. If O_L is free over the associated order A_{L/K}…

Let $L_n$ be a simplest cubic field with Galois group $G=\rm{Gal} (L_n/\mathbb Q)$. The associated order is denoted as ${\cal A}_{L_n/\mathbb Q}:= \{ x\in {\mathbb Q} [G] \, |\, x \cdot \cal{O}_{L_n} \subset {\cal O}_{L_n } \}$, where…

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…

Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$.…

We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-nX^2-(n+3)X-1$. Let $L_n=\mathbb Q(\rho_n)$ be a cyclic cubic field with Galois group…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that…

Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition for the ring of integers $\mathcal{O}_L$ to be free as a module over its associated order in the…

We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is an unramified extension of $ p…

Let $K$ be a field of prime characteristic $p$, $n>4 $ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Let $l$ be an odd prime…

Let L/K be an extension of absolutely abelian number fields of equal conductor, n. The image of the ring of integers of L under the trace map from L to K is an ideal in the ring of integers in K. We compute the absolute norm of this ideal…

We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study…

Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies…

We show that any Lambda-ring, in the sense of Riemann-Roch theory, which is finite etale over the rational numbers and has an integral model as a Lambda-ring is contained in a product of cyclotomic fields. In fact, we show that the category…

Given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n=2. As a…

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be…

Let $p$ be an odd prime. For field extensions $L/\mathbb{Q}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and…

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an…