Common slot lemmas in characteristic 2

TL;DR

This paper establishes and proves two new analogues of the common slot lemma for symbols in characteristic 2, extending known results from characteristic not 2.

Contribution

It introduces and proves two previously unconsidered analogues of the common slot lemma for symbols in characteristic 2.

Findings

Two new analogues of the common slot lemma are proven in characteristic 2.

The results extend the understanding of symbol relations in characteristic 2.

The paper completes the set of analogues for the symbols involved.

Abstract

The Hilbert symbol $(\cdot,\cdot )$ from characteristic $\neq 2$ has two analogues in characteristic $2$, $[\cdot,\cdot )$ and $((\cdot,\cdot ))$. In characteristic $\neq 2$ 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.