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Kernel Groups and nontrivial Galois module structure of imaginary quadratic fields | | Daniel R. Replogle
; | | Date: |
17 Dec 2001 | | Subject: | Number Theory | math.NT | | Abstract: | Let $K$ be an algebraic number field with ring of integers $Cal{O}_{K}$, $p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let $Lambda$ denote the order $Cal{O}_{K}[G].$ Let $Cl(Lambda)$ denote the locally free class group of $Lambda$ and $D(Lambda)$ the kernel group, the subgroup of $Cl(Lambda)$ consisting of classes that become trivial upon extension of scalars to the maximal order. If $p$ is unramified in $K$, then $D(Lambda) = T(Lambda)$, where $T(Lambda)$ is the Swan subgroup of $Cl(Lambda).$ This yields upper and lower bounds for $D(Lambda)$. Let $R(Lambda)$ denote the subgroup of $Cl(Lambda)$ consisting of those classes realizable as rings of integers, $Cal{O}_{L},$ where $L/K$ is a tame Galois extension with Galois group $Gal(L/K) cong G.$ We show under the hypotheses above that $T(Lambda)^{(p-1)/2} subseteq R(Lambda) cap D(Lambda) subseteq T(Lambda)$, which yields conditions for when $T(Lambda)=R(Lambda) cap D(Lambda)$ and bounds on $R(Lambda) cap D(Lambda)$. We carry out the computation for $K=Bbb{Q}(sqrt{-d}), d>0, d
eq 1$ or $3.$ In this way we exhibit primes $p$ for which these fields have tame Galois field extensions of degree $p$ with nontrivial Galois module structure. | | Source: | arXiv, math.NT/0201323 | | Services: | Forum | Review | PDF | Favorites | |
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