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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2409.16447 |
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| _version_ | 1866912894655922176 |
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| author | Chapman, Adam Krashen, Daniel McKinnie, Kelly |
| author_facet | Chapman, Adam Krashen, Daniel McKinnie, Kelly |
| contents | Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=ω\otimes β_1 \otimes \dots \otimes β_n$ in $H_{p^m}^{n+1}(F)$ is of exponent dividing $p^{m-1}$, then its symbol length in $H_{p^{m-1}}^{n+1}(F)$ is at most $p^n$. In the case $n=2$ we also prove that if $A= ω_1\otimes β_1+\cdots+ω_r\otimes β_r$ in $H_{p^{m}}^2(F)$ satisfies $\exp(A)|p^{m-1}$, then the symbol length of $A$ in $H_{p^{m-1}}^2(F)$ is at most $p^r+r-1$. We conclude by looking at the case $p=2$ and proving that if $A$ is a sum of two symbols in $H_{2^m}^{n+1}(F)$ and $\exp A |2^{m-1}$, then the symbol length of $A$ in $H_{2^{m-1}}^{n+1}(F)$ is at most $(2n+1)2^n$. Our results use norm conditions in characteristic $p$ in the same manner as Matrzi in his paper ``On the symbol length of symbols''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16447 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classes in $\mathrm H_{p^m}^{n+1}(F)$ of lower exponent Chapman, Adam Krashen, Daniel McKinnie, Kelly Rings and Algebras Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=ω\otimes β_1 \otimes \dots \otimes β_n$ in $H_{p^m}^{n+1}(F)$ is of exponent dividing $p^{m-1}$, then its symbol length in $H_{p^{m-1}}^{n+1}(F)$ is at most $p^n$. In the case $n=2$ we also prove that if $A= ω_1\otimes β_1+\cdots+ω_r\otimes β_r$ in $H_{p^{m}}^2(F)$ satisfies $\exp(A)|p^{m-1}$, then the symbol length of $A$ in $H_{p^{m-1}}^2(F)$ is at most $p^r+r-1$. We conclude by looking at the case $p=2$ and proving that if $A$ is a sum of two symbols in $H_{2^m}^{n+1}(F)$ and $\exp A |2^{m-1}$, then the symbol length of $A$ in $H_{2^{m-1}}^{n+1}(F)$ is at most $(2n+1)2^n$. Our results use norm conditions in characteristic $p$ in the same manner as Matrzi in his paper ``On the symbol length of symbols''. |
| title | Classes in $\mathrm H_{p^m}^{n+1}(F)$ of lower exponent |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2409.16447 |