具有2-幂次Frobenius-Galois群的Galois推广中的算术等价域

Pub Date : 2022-06-13 DOI:10.4153/S0008439522000388
Masanari Kida
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摘要

摘要设$F_{2^n}$是阶为$2^n$且阶为$2^2(2^n-1)$且具有$n\ge4$的Frobenius群。我们证明,如果$K/\mathbb{Q}$是Galois扩张,其Galois群同构于$F_{2^n}$,则存在$K/\math bb{Q}$的$\dfrac{2^{n-1}+(-1)^ n}{3}$中间域,阶为$4(2^n-1)$,使得它们在$\mathbb{Q}$上不共轭,而是在$\math bb{Q}$上算术等价。我们还给出了构造这些算术等价域的显式方法。
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Arithmetically equivalent fields in a Galois extension with Frobenius Galois group of 2-power degree
Abstract Let $F_{2^n}$ be the Frobenius group of degree $2^n$ and of order $2^n ( 2^n-1)$ with $n \ge 4$ . We show that if $K/\mathbb {Q} $ is a Galois extension whose Galois group is isomorphic to $F_{2^n}$ , then there are $\dfrac {2^{n-1} +(-1)^n }{3}$ intermediate fields of $K/\mathbb {Q} $ of degree $4 (2^n-1)$ such that they are not conjugate over $\mathbb {Q}$ but arithmetically equivalent over $\mathbb {Q}$ . We also give an explicit method to construct these arithmetically equivalent fields.
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