In this talk we will present a characterization for the defect of simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let $K^h$ be a henselization of a valued field $K$. Another relevant result we will discuss is that for every valuation $\mu^h$ on $K^h[x]$, with restriction $\mu$ on $K[x]$, the corresponding map $\mathcal G_\mu\longrightarrow\mathcal G_{\mu^h}$ of graded algebras is an isomorphism.
This is joint work with Enric Nart.
