  Fix an arbitrary prime $p$. Let $F$ be a field containing a primitive $p$-th
root of unity, with absolute Galois group $G_F$, and let $H^n$ denote its mod
$p$ cohomology group $H^n(G_F,\mathbb{Z}/p\mathbb{Z})$. The triple Massey
product of weight $(n,k,m)\in \mathbb{N}^3$ is a partially defined,
multi-valued function $\langle \cdot,\cdot,\cdot \rangle: H^n\times H^k\times
H^m\rightarrow H^{n+k+m-1}.$ %(in the mod-$p$ Galois cohomology) In this work
we prove that for an arbitrary prime $p$, any defined $3MP$ of weight
$(n,1,m)$, where the first and third entries are assumed to be symbols,
contains zero; and that for $p=2$ any defined $3MP$ of the weight $(1,k,1)$,
where the middle entry is a symbol, contains zero. Finally, we use the
description of the kernel of multiplication by a symbol to study general 3MP
where the middle slot is a symbol. The main tools we will be using is Lemma 4.1
concerning the the annihilator of cup product with an $H^1$ element, and
Theorem 5.2, generalizing a Theorem of Tignol on quaternion algebras with
trivial corestriction along a separable quadratic extension.
