Title: Analogues of the $p^n$th Hilbert symbol in characteristic $p$ (updated),
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 $x\mapsto x^p-x$, and $$((\cdot,\cdot ))_p:K/K^p\times K/K^p\to{}_p{\rm
Br}(K).$$
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.