Geodesic
Symbol length of classes in Milnor $K$-groups
Algebra i analiz, Tome 34 (2022) no. 4, pp. 214-221 Cet article a éte moissonné depuis la source Math-Net.Ru

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Given a field $F$, a positive integer $m$ and an integer $n\geq 2$, it is proved that the symbol length of classes in Milnor's $K$-groups $K_n F/2^m K_n F$ that are equivalent to single symbols under the embedding into $K_n F/2^{m+1} K_n F$ is at most $2^{n-1}$ under the assumption that $F \supseteq \mu_{2^{m+1}}$. Since $K_2 F/2^m K_2 F \cong {_{2^m}Br(F)}$ for $n=2$, this coincides with the upper bound of $2$ (proved by Tignol in $1983$) for the symbol length of central simple algebras of exponent $2^m$ that are Brauer equivalent to a single symbol algebra of degree $2^{m+1}$. The cases where the embedding into $K_n F/2^{m+1} K_n F$ is of symbol length $2$, $3$, and $4$ (the last when $n=2$) are also considered. The paper finishes with the study of the symbol length for classes in $K_3/3^m K_3 F$ whose embedding into $K_3 F/3^{m+1} K_3 F$ is one symbol when $F \supseteq \mu_{3^{m+1}}$.
Keywords: algebraic $K$-Theory, Milnor $K$-Theory, symmetric bilinear forms, quadratic forms, symbol length
Mots-clés : quaternion algebras. 
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     author = {A. Chapman},
     title = {Symbol length of classes in {Milnor} $K$-groups},
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     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2022_34_4_a6/}
}
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A. Chapman. Symbol length of classes in Milnor $K$-groups. Algebra i analiz, Tome 34 (2022) no. 4, pp. 214-221. http://geodesic.mathdoc.fr/item/AA_2022_34_4_a6/

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