purely inseparable extension

An algebraic extension $E\backslash supseteq\; F$ is a purely inseparable extension if and only if for every $\backslash alpha\backslash in\; E\backslash setminus\; F$, the minimal polynomial of $\backslash alpha$ over F is not a separable polynomial.Isaacs, p. 298 If F is any field, the trivial extension $F\backslash supseteq\; F$ is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.

Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If $E\backslash supseteq\; F$ is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:Isaacs, Theorem 19.10, p. 298

1. E is purely inseparable over F.
2. For each element $\backslash alpha\backslash in\; E$, there exists $n\backslash geq\; 0$ such that $\backslash alpha^\{p^n\}\backslash in\; F$.
3. Each element of E has minimal polynomial over F of the form $X^\{p^n\}-a$ for some integer $n\backslash geq\; 0$ and some element $a\backslash in\; F$.

It follows from the above equivalent characterizations that if $E=F[\backslash alpha]$ (for F a field of prime characteristic) such that $\backslash alpha^\{p^n\}\backslash in\; F$ for some integer $n\backslash geq\; 0$, then E is purely inseparable over F.Isaacs, Corollary 19.11, p. 298 (To see this, note that the set of all x such that $x^\{p^n\}\backslash in\; F$ for some $n\backslash geq\; 0$ forms a field; since this field contains both $\backslash alpha$ and F, it must be E, and by condition 2 above, $E\backslash supseteq\; F$ must be purely inseparable.)

If F is an imperfect field of prime characteristic p, choose $a\backslash in\; F$ such that a is not a pth power in F, and let f(X) = X^p − a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose $\backslash alpha$ with $f(\backslash alpha)=0$. In particular, $\backslash alpha^\{p\}=a$ and by the property stated in the paragraph directly above, it follows that $F[\backslash alpha]\backslash supseteq\; F$ is a non-trivial purely inseparable extension (in fact, $E=F[\backslash alpha]$, and so $E\backslash supseteq\; F$ is automatically a purely inseparable extension).Isaacs, p. 299

Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K(V) is a purely inseparable extension over the subfield K(V)^p of pth powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by p on an elliptic curve over a finite field of characteristic p.

• If the characteristic of a field F is a (non-zero) prime number p, and if $E\backslash supseteq\; F$ is a purely inseparable extension, then if $F\backslash subseteq\; K\backslash subseteq\; E$, K is purely inseparable over F and E is purely inseparable over K. Furthermore, if [E : F] is finite, then it is a power of p, the characteristic of F.Isaacs, Corollary 19.12, p. 299
• Conversely, if $F\backslash subseteq\; K\backslash subseteq\; E$ is such that $F\backslash subseteq\; K$ and $K\backslash subseteq\; E$ are purely inseparable extensions, then E is purely inseparable over F.Isaacs, Corollary 19.13, p. 300
• An algebraic extension $E\backslash supseteq\; F$ is an inseparable extension if and only if there is some $\backslash alpha\backslash in\; E\backslash setminus\; F$ such that the minimal polynomial of $\backslash alpha$ over F is not a separable polynomial (i.e., an algebraic extension is inseparable if and only if it is not separable; note, however, that an inseparable extension is not the same thing as a purely inseparable extension). If $E\backslash supseteq\; F$ is a finite degree non-trivial inseparable extension, then [E : F] is necessarily divisible by the characteristic of F.Isaacs, Corollary 19.16, p. 301
• If $E\backslash supseteq\; F$ is a finite degree normal extension, and if $K=\backslash mbox\{Fix\}(\backslash mbox\{Gal\}(E/F))$, then K is purely inseparable over F and E is separable over K.Isaacs, Theorem 19.18, p. 301

{{harvs|txt|last=Jacobson|year1=1937|year2=1944}} introduced a variation of Galois theory for purely inseparable extensions of exponent 1, where the Galois groups of field automorphisms in Galois theory are replaced by restricted Lie algebras of derivations. The simplest case is for finite index purely inseparable extensions K⊆L of exponent at most 1 (meaning that the pth power of every element of L is in K). In this case the Lie algebra of K-derivations of L is a restricted Lie algebra that is also a vector space of dimension n over L, where [L:K] = pⁿ, and the intermediate fields in L containing K correspond to the restricted Lie subalgebras of this Lie algebra that are vector spaces over L. Although the Lie algebra of derivations is a vector space over L, it is not in general a Lie algebra over L, but is a Lie algebra over K of dimension n[L:K] = npⁿ.

A purely inseparable extension is called a modular extension if it is a tensor product of simple extensions, so in particular every extension of exponent 1 is modular, but there are non-modular extensions of exponent 2 {{harv|Weisfeld|1965}}.

{{harvtxt|Sweedler|1968}} and {{harvtxt|Gerstenhaber|Zaromp|1970}} gave an extension of the Galois correspondence to modular purely inseparable extensions, where derivations are replaced by higher derivations.

• Jacobson–Bourbaki theorem

{{reflist}}

• {{Citation | last1=Gerstenhaber | first1=Murray |authorlink1=Murray Gerstenhaber| last2=Zaromp | first2=Avigdor | title=On the Galois theory of purely inseparable field extensions | doi=10.1090/S0002-9904-1970-12535-6 |mr=0266904 | year=1970 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=76 | issue=5 | pages=1011–1014| doi-access=free }}
• {{citation

| first = I. Martin |last=Isaacs

| year = 1993

| title = Algebra, a graduate course

| edition = 1st

| publisher = Brooks/Cole Publishing Company

| isbn = 0-534-19002-2

}}

• {{Citation | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Abstract Derivation and Lie Algebras | jstor=1989656 | publisher=American Mathematical Society | location=Providence, R.I. | year=1937 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=42 | issue=2 | pages=206–224 | doi=10.2307/1989656| doi-access=free }}
• {{Citation | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Galois theory of purely inseparable fields of exponent one | jstor= 2371772 |mr=0011079 | year=1944 | journal=American Journal of Mathematics | issn=0002-9327 | volume=66 | issue=4 | pages=645–648 | doi=10.2307/2371772}}
• {{Citation | last1=Sweedler | first1=Moss Eisenberg | author1-link= Moss Sweedler |title=Structure of inseparable extensions | jstor=1970711 |mr=0223343 | year=1968 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=87 | issue=3 | pages=401–410 | doi=10.2307/1970711}}
• {{Citation | last1=Weisfeld | first1=Morris | title=Purely inseparable extensions and higher derivations | jstor=1994126 |mr=0191895 | year=1965 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=116 | pages=435–449 | doi=10.2307/1994126| doi-access=free }}

Category:Field (mathematics)