Sums of two symbols in $K_2(F)/2K_2(F)$ in characteristic two
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917416008679424 |
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| author | Barry, Demba Chapman, Adam Laghribi, Ahmed |
| author_facet | Barry, Demba Chapman, Adam Laghribi, Ahmed |
| contents | In this paper, study sums $A=\{a,b\}_2+\{c,d\}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\operatorname{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{α,β\}_2+\{γ,δ\}_2$ by a finite sequence of small steps when $A \equiv B$. We use this lemma to prove that $\{a,b,c,d\}_2 \in K_4(F)/2K_4(F)$ is a well-defined invariant of $A$, and that this invariant is trivial if and only if $A$ is congruent to a single symbol in $K_2(F)/4K_2(F)$. We also bound the symbol length of $C$ in $K_2(F)/2^m K_2(F)$ from above when $C$ is the sum of up to four symbols in $K_2(F)/2^{m+1}K_2(F)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15928 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sums of two symbols in $K_2(F)/2K_2(F)$ in characteristic two Barry, Demba Chapman, Adam Laghribi, Ahmed K-Theory and Homology Number Theory 11E81, 11E04, 19D45 In this paper, study sums $A=\{a,b\}_2+\{c,d\}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\operatorname{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{α,β\}_2+\{γ,δ\}_2$ by a finite sequence of small steps when $A \equiv B$. We use this lemma to prove that $\{a,b,c,d\}_2 \in K_4(F)/2K_4(F)$ is a well-defined invariant of $A$, and that this invariant is trivial if and only if $A$ is congruent to a single symbol in $K_2(F)/4K_2(F)$. We also bound the symbol length of $C$ in $K_2(F)/2^m K_2(F)$ from above when $C$ is the sum of up to four symbols in $K_2(F)/2^{m+1}K_2(F)$. |
| title | Sums of two symbols in $K_2(F)/2K_2(F)$ in characteristic two |
| topic | K-Theory and Homology Number Theory 11E81, 11E04, 19D45 |
| url | https://arxiv.org/abs/2604.15928 |