Separable extension

2020 Mathematics Subject Classification: Primary: 12Fxx [MSN][ZBL]

A separable extension of a field $k$ is an extension $K/k$ such that for some natural number $n$ the fields $K$ and $k^{p^{-n}}$ are linearly disjoint over $k$ (see Linearly-disjoint extensions). An extension that is not separable is called inseparable. Here $p$ is the characteristic of $k$. In characteristic 0 all extensions are separable.

In what follows only algebraic extensions will be considered (for transcendental separable extensions see Transcendental extension). A finite extension is separable if and only if the trace mapping ${\mathrm Tr} : K\to k$ is a non-zero function. An algebraic extension is separable if any finite subextension is separable.

The separable extensions form a distinguished class of extensions, that is, in a tower of fields $L\supset K\supset k$ the extension $L/k$ is separable if and only if $L/K$ and $K/k$ are separable; if $K_1/k$ and $K_2/k$ are separable extensions, then so is $K_1K_2/k$; for a separable extension $K/k$ and an arbitrary extension $L/k$ the extension $KL/L$ is again separable. An extension $K/k$ is separable if and only if it admits an imbedding in a Galois extension $L/k$. In this case, the number of different $k$-isomorphisms of $K$ into $L$ is the same as the degree $[K:k]$ for a finite extension $K/k$. Any finite separable extension is simple.

A polynomial $f\in k[x]$ is called separable over $k$ if none of its irreducible factors has a multiple root in an algebraic closure of $k$. An algebraic element $\def\a{\alpha}$ is called separable (over $k$) if it is a root of a polynomial that is separable over $k$. Otherwise $\a$ is called inseparable. An element $\a$ is called purely inseparable over $k$ if $\a^{p^n}\in k$ for some $n$. An irreducible polynomial $f(x)$ is inseparable if and only if its derivative $f'(x)$ is identically zero (this is possible only for $k$ of characteristic $p>0$ and $f(x)=f_1(x^p)$). An arbitrary irreducible polynomial $f(x)$ can be uniquely represented in the form $f(x)=g(x^{p^e})$, where $g(x)$ is a separable polynomial. The degree of $g(x)$ and the number $e$ are called, respectively, the reduced degree and the index of $f(x)$.

Let $L/k$ be an arbitrary algebraic extension. The elements of the field $L$ that are separable over $k$ form a field, $K$, which is the maximal separable extension of $k$ contained in $L$. The field $K$ is called the separable closure of $k$ in $L$. The degree $[K:k]$ is called the separable degree of $L/k$, and the degree $[L:K]$ the inseparable degree, or the degree of inseparability. The inseparable degree is equal to some power of the number $p=\mathrm{char\;} k$. If $K=k$, then $k$ is said to be separably closed in $L$. In this case the extension $L/k$ is called purely inseparable. An extension $K/k$ is purely inseparable if and only if $$K\subset k^{p^\infty} = \bigcup_n\; k^{p^{-n}},$$ that is, if any element of $K$ is purely inseparable over $k$. The purely inseparable extensions of a field $k$ form a distinguished class of extensions. If an extension $K/k$ is both separable and purely inseparable, then $K=k$. For references see Extension of a field.