On the unramified common divisor of discriminants of integers in a normal extension (Q2709972)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the unramified common divisor of discriminants of integers in a normal extension |
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On the unramified common divisor of discriminants of integers in a normal extension (English)
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5 September 2001
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algebraic numbers
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rings of algebraic integers
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class numbers
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class groups
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discriminants
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0.8902553
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0.87676835
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0.87519884
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0.8729849
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0.8722751
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0.87158126
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0.8713577
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Let \(F\) be an algebraic number field of a finite degree, and \(K\) be an extension over \(F\) of a finite degree. It is known that the greatest common divisor of the discriminants of integers of \(K\) with respect to \(K/F\) is not always equal to the discriminant of \(K/F\). In this paper, we assume that \(K/F\) is normal, and give a simple criterion to determine for a prime ideal of \(F\), which is unramified in \(K/F\), to divide the greatest common divisor of the discriminants of integers of \(K\) with respect to \(K/F\). The result is obtained only by an elementary investigation of conjugates of integers.
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