# The most general splitting of a field extension

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.)
• By "splitting" an extension do you mean picking out an intermediate extension or refining it into a tower of extensions? You're essentially asking about the structure of the lattice of intermediate fields. What do you mean by "more precise" splittings for local fields? – anon Jan 30 '15 at 17:02
• @anon This is what I mean by splitting, yes. Like first a separable extension, and then another of another type after it etc, for instance, plus the fact that we could have structure theorems for separable extensions etc... By more precise for local fields I am referring to the one with ramification. – EricFlorentNoube Jan 30 '15 at 17:10
• Eric, you probably know this, but just to make sure: the intermediate field $E$ is not uniquely determined by the pair $L/K$ - only the transcendence degree of $E/K$ is fixed. About your question. I'm not sure I understand what kind of splitting you are looking for. In characteristic $p$ an extension can be done in two steps - one separable, the other purely inseparable. Is that what you are looking for? Extensions of local fields (may be we need Galois, too late here for me to think straight) can be similarly done with an unramified step and a fully ramified step. – Jyrki Lahtonen Jan 30 '15 at 23:15
• @JyrkiLahtonen Yes Jyrki, I know that. For the splitting you're mentionning in char. $p$ (and in arbitrary dimension), it is exactly the kind of splitting I am looking in general, for a general extension $L/K$, in arbitrary char. – EricFlorentNoube Jan 30 '15 at 23:21