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- W2789919732 abstract "The $p$th degree Hilbert symbol $(cdot,cdot )_p:K^times/K^{times p}times K^times/K^{times p}to{}_p{rm Br}(K)$ from characteristic $neq p$ has two analogues in characteristic $p$, $$[cdot,cdot )_p:K/wp (K)times K^times/K^{times p}to{}_p{rm Br}(K),$$ where $wp$ is the Artin-Schreier map $xmapsto x^p-x$, and $$((cdot,cdot ))_p:K/K^ptimes K/K^pto{}_p{rm Br}(K).$$ The symbol $[cdot,cdot )_p$ generalizes to an analogue of $(cdot,cdot )_{p^n}$ via the Witt vectors, $$[cdot,cdot )_{p^n}:W_n(K)/wp (W_n(K))times K^times/K^{times p^n}to{}_{p^n}{rm Br}(K).$$ Here $W_n(K)$ is the truncation of length $n$ of the ring of $p$-typical Witt wectors, i.e. $W_{{1,p,ldots,p^{n-1}}}(K)$. In this paper we construct similar generalizations for $((cdot,cdot ))_p$. Our construction involves Witt vectors and Weyl algebras. In the process we obtain a new kind of Weyl algebras in characteristic $p$, with many interesting properties. The symbols we introduce, $((cdot,cdot ))_{p^n}$ and, more generally, $((cdot,cdot ))_{p^m,p^n}$, which here are defined in terms of central simple algebras, coincide with the homonymous symbols we introduced in [arXiv:1711.00980] in terms of the symbols $[cdot,cdot )_{p^n}$. This will be proved in a future paper. In the present paper we only introduce the symbols and we prove that they have the same properties with the symbols from [arXiv:1711.00980]. These properies are enough to obtain the representation theorem for ${}_{p^n}{rm Br}(K)$ from [arXiv:1711.00980], Theorem 4.10." @default.
- W2789919732 created "2018-03-29" @default.
- W2789919732 creator A5035792267 @default.
- W2789919732 date "2017-07-03" @default.
- W2789919732 modified "2023-09-27" @default.
- W2789919732 title "Analogues of the $p^n$th Hilbert symbol in characteristic $p$ (updated)" @default.
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- W2789919732 hasPublicationYear "2017" @default.
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