On defectless unibranched simple extensions, complete distinguished chains and certain stability results

Let $(K,v)$ be a valued field. Take an extension of $v$ to a fixed algebraic closure $L$ of $K$. In this paper we show that an element $a\in L$ admits a complete distinguished chain over $K$ if and only if the extension $(K(a)|K,v)$ is defectless and unibranched. This characterization generalizes the known result in the henselian case. In particular, our result shows that if $a$ admits a complete distinguished chain over $K$, then it also admits one over the henselization; however, the converse may not be true. The main tool employed in our analysis is the stability of the $j$-invariant associated to a valuation transcendental extension under passage to the henselization. We also explore the stability of defectless simple extensions in the following sense: let $(K(X)|K,w)$ be a valuation transcendental extension with a pair of definition $(b,\gamma)$. Assume that either $(K(b)|K,v)$ is a defectless extension, or that $f(X)$ is a key polynomial for $w$ over $K$, where $f(X)$ is the minimal polynomial of $b$ over $K$. We show that then the extension $(K(b,X)|K(X),w)$ is defectless. In particular, the extension $(K(b,X)|K(X),w)$ is always defectless whenever $(b,\gamma)$ is a minimal pair of definition for $w$ over $K$.