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Another application of Milnor K-groups:

1. The following are equivalent:

${a_1, \ldots, a_n} = 0 \in K^M_n(K)/2$

$\langle\kern-0.2em\langle{a_1, \ldots, a_n}\rangle\kern-0.2em\rangle$ (Pfister form) is totally hyperbolic

$\langle\kern-0.2em\langle{a_1, \ldots, a_n}\rangle\kern-0.2em\rangle$ is isotropic

$a_n$ is represented by $\langle\kern-0.2em\langle{a_1, \ldots, a_{n-1}\rangle\kern-0.2em}\rangle$

1. higher local class field theory: The class formation of an $n$-dimensional local field is $K^M_n(K)$. http://www.emis.de/journals/GT/ftp/main/m3/
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Another application of Milnor K-groups:

The following are equivalent:

${a_1, \ldots, a_n} = 0 \in K^M_n(K)/2$

$\langle\kern-0.2em\langle{a_1, \ldots, a_n}\rangle\kern-0.2em\rangle$ (Pfister form) is totally hyperbolic

$\langle\kern-0.2em\langle{a_1, \ldots, a_n}\rangle\kern-0.2em\rangle$ is isotropic

$a_n$ is represented by $\langle\kern-0.2em\langle{a_1, \ldots, a_{n-1}\rangle\kern-0.2em}\rangle$