Abstract: Let $K|\bQ_p(\zeta)$ be a finite extension where $p$ is a fixed prime number and $\zeta$ is a primitive $p$-th root of 1. The filtration $(U_n)_{n>0}$ on $K^{\times}$ by units of various levels induces a filtration on the $\mathbb{F}_p$-space $\overline{K^{\times}}=K^{\times}/K^{\times p}$ denoted by $(\overline{U}_n)_{n>0}$. Let $M=K(\sqrt[p]{K^\times})$ be the maximal elementary abelian $p$-extension of $K$, and let $G =\Gal(M|K)$, endowed with the ramification filtration $(G^u)_{u \in [-1,+\infty[ }$ in the upper numbering. I will explain the relationship between $(G^u)_u$ and $(\overline{U}_n)_{n>0}$. This relationship allows us to compute the discriminant of any elementary abelian $p$-extension of local fields, without invoking class field theory, etc. This talk will be about general background materials rather than new results.