Let $X/k$ be a smooth, projective curve over $k$ and let $L/k$ be a finite extension of fields, where $k$ is a finite extension of $\mathbb{Q}_p$, $p \not=2$. Suppose $k(X)$ contains no elements algebraic over $k$ other than $k$ itself. For any field $K$ let $Br(K)$ denote the Brauer group of $K$.
Now assume we are given $(a,b)$, a non-trivial cyclic algebra of degree 2 in $Br(k)$ which is still non-trivial in $Br(k(X))$.
If $(a,b)$ is also non-trivial in $Br(L)$, is it true that $(a,b)$ is non-trivial in $Br(L(X))$?
It seems like the answer morally should be 'Yes', since I am simply extending the base-field with elements which don't help me solve the equation. However, I cannot see a rigorous proof of this, and I am now unsure if there might not be a counter-example. An answer either direction would be much appreciated.