It is proved that, for a (henselian) local field K and for a fixed Lubin-Tate splitting f over K, the metabelian local Artin map (?, K)f: B(K, f) \tilde{\rightarrow} Gal (K(ab)2 / K) satisfies the Galois conjugation law (\tilde{s}+(a), s (K))\tilde{s}f\tilde{s}-1 = \tilde{s}|K(ab)2 (a, K)f\tilde{s}-1|\tilde{s}(K(ab)2) for any a \in B(K, f), and for any embedding s : K \hookrightarrow Ksep, where \tilde{s} \in Aut (Ksep) is a fixed extension to Ksep of the embedding s : K \hookrightarrow Ksep.
📌 3 Key Takeaways
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Objective & Hypothesis:It is proved that, for a (henselian) local field K and for a fixed Lubin-Tate splitting f over K, the metabelian local Artin map (?, K)f: B(K, f) \tilde{\rightarrow} Gal (K(ab)2 / K) satisfies the Galois conjugation law (\tilde{s}+(a), s (K))\tilde{s}f\tilde{s}-1 = \tilde{s}|K(ab)2 (a, K)f\tilde{s}-1|\tilde{s}(K(ab)2) for any a \in B(K, f), and for any embedding s : K \hookrightarrow Ksep, where \tilde{s} \in Aut (Ksep) is a fixed extension to Ksep of the embedding s : K \hookrightarrow Ksep.
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Key Findings & Data:It is proved that, for a (henselian) local field K and for a fixed Lubin-Tate splitting f over K, the metabelian local Artin map (?, K)f: B(K, f) \tilde{\rightarrow} Gal (K(ab)2 / K) satisfies the Galois conjugation law (\tilde{s}+(a), s (K))\tilde{s}f\tilde{s}-1 = \tilde{s}|K(ab)2 (a, K)f\tilde{s}-1|\tilde{s}(K(ab)2) for any a \in B(K, f), and for any embedding s : K \hookrightarrow Ksep, where \tilde{s} \in Aut (Ksep) is a fixed extension to Ksep of the embedding s : K \hookrightarrow Ksep.
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Conclusion & Impact:It is proved that, for a (henselian) local field K and for a fixed Lubin-Tate splitting f over K, the metabelian local Artin map (?, K)f: B(K, f) \tilde{\rightarrow} Gal (K(ab)2 / K) satisfies the Galois conjugation law (\tilde{s}+(a), s (K))\tilde{s}f\tilde{s}-1 = \tilde{s}|K(ab)2 (a, K)f\tilde{s}-1|\tilde{s}(K(ab)2) for any a \in B(K, f), and for any embedding s : K \hookrightarrow Ksep, where \tilde{s} \in Aut (Ksep) is a fixed extension to Ksep of the embedding s : K \hookrightarrow Ksep.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:local fields · metabelian extensions · metabelian local Artin map
Abstract
It is proved that, for a (henselian) local field K and for a fixed Lubin-Tate splitting f over K, the metabelian local Artin map (?, K)f: B(K, f) \tilde{\rightarrow} Gal (K(ab)2 / K) satisfies the Galois conjugation law (\tilde{s}+(a), s (K))\tilde{s}f\tilde{s}-1 = \tilde{s}|K(ab)2 (a, K)f\tilde{s}-1|\tilde{s}(K(ab)2) for any a \in B(K, f), and for any embedding s : K \hookrightarrow Ksep, where \tilde{s} \in Aut (Ksep) is a fixed extension to Ksep of the embedding s : K \hookrightarrow Ksep.
Özet
It is proved that, for a (henselian) local field K and for a fixed Lubin-Tate splitting f over K, the metabelian local Artin map (?, K)f: B(K, f) \tilde{\rightarrow} Gal (K(ab)2 / K) satisfies the Galois conjugation law (\tilde{s}+(a), s (K))\tilde{s}f\tilde{s}-1 = \tilde{s}|K(ab)2 (a, K)f\tilde{s}-1|\tilde{s}(K(ab)2) for any a \in B(K, f), and for any embedding s : K \hookrightarrow Ksep, where \tilde{s} \in Aut (Ksep) is a fixed extension to Ksep of the embedding s : K \hookrightarrow Ksep.
İKEDA, K. İ. (2022). On the Metabelian Local Artin Map I: Galois Conjugation Law. Turkish Journal of Mathematics, Cilt: 24 - Sayı: 1, 25–58. https://ajindex.com/articles/608928