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at 12:15pm
in MH1060
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(University of Calgary)
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Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}\geq 1$ for any algebraic integer $\alpha$ of degree $d$, where we label its Galois conjugates as $\alpha_0,\ldots,\alpha_{d-1}$ with
$\vert\alpha_0\vert\geq \vert\alpha_1\vert\geq\cdots \geq \vert\alpha_{d-1}\vert$. First, we give an explicit description of $E_{k,d}$ as a polytope with $2^k$ vertices. Then we prove that for $d>3k$, for every $(c_1,\ldots,c_k)\in E_{k,d}$ and for every $\alpha$ that is not a root of unity, the strict inequality $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}>1$
holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $\alpha$.
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at 12:15pm
in MH1060
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(University of Lethbridge)
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The Polya group $Po(K)$ of a Galois number field $K$ coincides with the subgroup of the ideal class group $Cl(K)$ of $K$ consisting of all strongly ambiguous ideal classes. We prove that there are only finitely many imaginary abelian number fields $K$ whose `Polya index' $\left[Cl(K):Po(K)\right]$ is a fixed integer. Accordingly, under GRH, we completely classify all imaginary quadratic fields with the Polya indices 1 and 2. Also, we unconditionally classify all imaginary biquadratic and imaginary tri-quadratic fields with the Polya index 1. In another direction, we classify all real quadratic fields $K$ of extended R-D type (with possibly only one more field $K$) for which $Po(K)=Cl(K)$. Our result generalizes Kazuhiro's classification of all real quadratic fields of narrow R-D type whose narrow genus numbers are equal to their narrow class numbers. This is a joint work with Amir Akbary (University of Lethbridge).
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