Take $L/K$ an extension of the field $K$. I have questions on how we can "split" the extension $L/K$.
- (1) One knows that $L$ has a transcendence basis $(x_i)_{i\in I}$ over $L$, so that if $E := K((x_i)_{i\in I})$ the field $E$ is (isomorphic over $K$ to) a field of rational functions in the formal variables $(X_i)_{i\in I}$ and that $L/E$ is algebraic. So that $L/K$ splits into $E/K$ "purely" transcendental and well-explicited and $E/L$ algebraic. How can we split $L/E$ now ? That, what is the more general way to split an algebraic extension (that is not necessarily finite, it could be, but it couldn't) ? (I guess it epends of characteristic of $k$, and on if $K$ is finite or not.)
- (2) Are there any other forms of general splitting of $L/K$ where we don't start as in (1) ?
- (3) Local fields have even more precise splitting, what is the more general splitting for a general extension $L/K$ of a local field then ? (Will depend on the characteristics of $K$ and its residue field I guess.)
- (4) Are there any other "classes" of fields different than local fields for which we have nice splittings ? (I am btw not interested for now in differential fields or ordered fields.)