Common slot lemmas in characteristic 2

Abstract

The Hilbert symbol $(\cdot ,\cdot )$ from characteristic $\ne 2$ has two analogues in characteristic $2$, $[\cdot ,\cdot )$ and $((\cdot ,\cdot ))$. In characteristic $\ne 2$ we have a well known result called the common slot lemma or the chain lemma. It involves the symbol $(\cdot ,\cdot )$. This result has three analogues in characteristic $2$. Two of these analogues involve the symbol $[\cdot ,\cdot )$ and one involves $((\cdot ,\cdot ))$. So far only one of these analogues has been considered. In this paper we state and prove the remaining two analogues.