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- W657821727 abstract "Let $F$ be an algebraic number field of a finite degree, and let $K$ be an extension of $F$ of a finite degree. Denote by $delta(K/F)$ the greatest common divisor of the discriminants of integers of $K$ with respect to $K/F$. Then, $delta(K/F)$ is divisible by the discriminant $d(K/F)$ of $K/F$. Let $mathfrak{p}$ be an arbitrary prime ideal of $F$, let $mathfrak{p}=mathfrak{q}_{1}^{e_1} mathfrak{q}_{2}^{e_2}cdots mathfrak{q}_{g}^{e_g}$ be the decomposition of $mathfrak{p}$ in $K$ into primes, and let $f_i$ be the degree of $mathfrak{q}_i$. The set of indices ${1, 2, ldots, g}$ is then divided into the union of maximal subsets $I$ such that $f_i = f_j$ whenever $i$ and $j$ belong to a common $I$. We write $f_I$ instead of $f_i$ for $i in I$, and denote by $g_I$ the number of elements in $I$. Put on the other hand $c(I)= sum_{d|f_{I}}mu(f_{I}/d)Nmathfrak{p}^{d}$, where $mu$ is the Möbius function. Then, $mathfrak{p}$ divides $delta(K/F)d(K/F)^{-1}$ if and only if there exists an $I$ such that $c(I) lt f_{I}g_{I}$." @default.
- W657821727 created "2016-06-24" @default.
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- W657821727 date "2002-06-01" @default.
- W657821727 modified "2023-10-18" @default.
- W657821727 title "On the common divisor of discriminants of integers" @default.
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- W657821727 doi "https://doi.org/10.21099/tkbjm/1496164382" @default.
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