

\titleofpresentation{From abstract co-Galois theory to field theoretic
co-Galois theory}

\presenter{\c Serban A. Basarab}  

An abstract (topological and group theoretic) approach of 
the field theoretic co-Galois theory is initiated in the recent 
papers \cite{Basarab1} and \cite{Basarab3}. Given a profinite group 
$\Gamma$ and a quasi-cyclic discrete group $A$ on which $\Gamma$ 
acts continuously, several classes of closed subgroups of $\Gamma$ 
and related classes of subgroups of the cocycle group $Z^1(\Gamma, A)$, 
called \emph{radical, hereditarily radical, Kneser, hereditarily Kneser} 
and \emph{co-Galois}, are defined, investigated and characterized. 

A more general frame for the abstract co-Galois theory is the
object of a recent project of the author \cite{Basarab4},
\cite{Basarab5}. In this more general setting one considers a
continuous action of a profinite group $\,\Gamma$ on a (not
necessarily commutative) profinite group $G$ together with a
continuous $1$-cocycle $\eta : \Gamma \longrightarrow G$ with
the property that the profinite group $G$ is topologically 
generated by the image $\eta(\Gamma)$. The main object of
study is the natural Galois connection between the topological
lattice of all closed subgroups of $\Gamma$ and the topological
lattice of all $\Gamma$-invariant closed normal subgroups of $G$.
The theory developed in \cite{Basarab1}, \cite {Basarab3}, which we 
may call \emph{cyclotomic abstract co-Galois theory} is the
first stage of this more general theory, where $G := {\rm Hom}
(Z^1(\Gamma, A), A)$ is the Pontryagin dual of the discrete 
Abelian torsion group $Z^1(\Gamma, A)$ with the action induced 
by the action of $\,\Gamma$ on the quasi-cyclic group $A$, while 
the cocycle $\eta : \Gamma \longrightarrow G$ is defined by 
$\eta(\gamma)(\alpha) = \alpha(\gamma)$ for $\,\alpha \in 
Z^1(\Gamma, A), \gamma \in \Gamma$.

The main results of the field theoretic co-Galois theory
are easily obtained from their cyclotomic abstract
co-Galois theoretic versions. Moreover, new results in the 
field theoretic co-Galois theory are derived from specific 
results of the cyclotomic co-Galois theory \cite{Basarab5}. 
On the other hand, the more general frame for the abstract 
co-Galois theory provides a suitable extension of the 
frame of the classical field theoretic co-Galois theory.   

We illustrate the introductory remarks above with an
extension of the abstract Kneser criterion 
\cite[Theorem 1.20]{Basarab1}, an application of a result 
concerning the so called \emph{quasi-co-Galois subgroups} 
to the description of the \emph{minimal non-co-Galois 
hereditarily Kneser algebraic separable field extensions} 
and an example of \emph{strongly co-Galois
algebraic separable field extensions} related to the
model theory of Henselian fields and $p$-adically closed
fields.   

\subsection*{Surjective cocycles and minimal non-Kneser structures}

The next result is an extension of \cite[Theorem 1.20]{Basarab1}.

\begin{proposition} \emph{\cite[Proposition 6.23]{Basarab4}
(Abstract Kneser Criterion)} Let $G$ be a profinite
$\Gamma$-operator group and $\eta \in Z^1(\Gamma, G)$
be a \emph{generating cocycle}, \emph{i.e.}\ the profinite group $G$
is topologically generated by $\eta(\Gamma)$. The following
assertions are equivalent for any $\Gamma$-invariant
closed normal subgroup $\bf a$ of $G$.
\begin{enumerate}
\item[\rm (1)] The induced cocycle $\eta_{\bf a} \in 
Z^1(\Gamma, G/{\bf a})$ is surjective.
\item[\rm (2)] ${\bf a} \not \subseteq {\bf m}$ for
all $\Gamma$-invariant open normal subgroups ${\bf m}$ of
$G$ wich are maximal with the property that the induced cocycle
$\eta_{\bf m} \in Z^1(\Gamma, G/{\bf m})$ is not surjective.
\end{enumerate}
\end{proposition}

{The following class of finite algebraic structures arises
naturally from the abstract Kneser criterion above.}

\begin{definition}
A triple $(\Gamma, G, \eta)$ consisting of a finite
group $\Gamma$, a finite $\Gamma$-operator group $G$ and a generating 
cocycle $\eta \in Z^1(\Gamma, G)$, with $\Delta := {\rm Ker}(\eta)$, 
is called a \emph{minimal non-Kneser structure} if the following 
conditions are satisfied :
\begin{enumerate}
\item[\rm (1)] $\eta(\Gamma) \neq G$,
\item[\rm (2)] for every $\Gamma$-invariant normal subgroup 
${\bf a} \neq \{ 1 \},\,(\Gamma : \eta^{- 1}({\bf a})) = 
(G : {\bf a})$, and  
\item[\rm (3)] the cocycle $\eta$ is \emph{normalised}, \emph{i.e.}\
$\displaystyle\bigcap_{\gamma \in \Gamma} \gamma \Delta \gamma^{- 1} = 
{\rm Fix}_\Gamma(G) \cap \Delta = \{ 1 \}$.
\end{enumerate}
  \end{definition}

\begin{problem*}
Classify up to isomorphism the
minimal non-Kneser structures.
\end{problem*}

A proper subclass of minimal non-Kneser structures is
provided by the next obvious lemma.

\begin{lemma} \emph{\cite[Lemma 7.3]{Basarab4}}
Let $\eta \in Z^1(\Gamma, G)$ be a normalised generating
cocycle such that $\eta(\Gamma) \neq G$. Then 
$(\Gamma, G, \eta)$ is a minimal non-Kneser structure
whenever the $\Gamma$-operator group $G$ is simple, \emph{i.e.}\
$\{ 1 \}$ and $G \neq \{ 1 \}$ are its only $\Gamma$-invariant
normal subgroups.
\end{lemma}

{The next results provide some classes of minimal non-Kneser 
structures including the very simple ones which occur in 
\cite[Lemma 1.18 and Theorem 1.20]{Basarab1}.}

\begin{lemma} \emph{\cite[Lemma 7.7]{Basarab4}}
Let $(\Gamma, G, \eta)$ be a minimal non-Kneser structure.
Assume that $\Gamma$ and $G$ are Abelian. The following
assertions hold.
\begin{enumerate}
\item[\rm (1)] $G$ is an Abelian $p$-group for some
prime number $p,\,\Gamma$ acts faithfully on $G$ and 
the generating cocycle $\eta$ is injective. Let 
$p^n, n \geq 1$, be the exponent of $G$.
\item[\rm (2)] There exists a unique minimal non-zero
$\Gamma$-submodule of $G$.
\item[\rm (3)] The image $R$ of the canonical ring
morphism $\mathbb{Z}/p^n \mathbb{Z}\,[\Gamma] 
\longrightarrow {\rm End}(G)$ is a finite
commutative local ring of characteristic $p^n$, and
$\Gamma$, identified with a subgroup of $R^\ast$,
generates $R$ as $\mathbb{Z}/p^n\mathbb{Z}$-module, 
in particular as ring.
\end{enumerate}
\end{lemma}

\begin{proposition} \emph{\cite[Proposition 7.8]{Basarab4}}
Let $\eta \in Z^1(\Gamma, G)$, where $\Gamma$ and $G$
are Abelian and finite, $G$ is a $p$-group for
some prime number $p$, while $\Gamma$ is not a $p$-group. 
Then, the following assertions are equivalent.
\begin{enumerate}
\item[\rm (1)] $\,(\Gamma, G, \eta)$ is a minimal
non-Kneser structure.
\item[\rm (2)] The local ring $R \subseteq {\rm End}(G)$ is 
a finite field $k \cong \mathbb{F}_q, q = p^f, f \geq 1$
for $p \neq 2,\,f \geq 2$ for $p = 2,\,G$ is a 
one-dimensional $k$-vector space identified with
$k^+$, the group $\Gamma$, identified with a subgroup of the
multiplicative group $k^\ast$, is cyclic of order
$1\neq r\,|\,(q - 1)$ such that $f$ is the order
of $p\,{\rm mod}\,r \in (\mathbb{Z}/r\mathbb{Z})^\ast$, and 
$\eta \in Z^1(\Gamma, k^+) = B^1(\Gamma, k^+) \cong
(\mathbb{Z}/p\mathbb{Z})^f$ is, up to multiplication by elements
in $\,k^\ast$, the coboundary $\,u \in \Gamma \mapsto
u - 1 \in k^+$.   
\end{enumerate}
\end{proposition}

\begin{proposition} \emph{\cite[Proposition 7.9]{Basarab4}}
Let $\eta \in Z^1(\Gamma, G)$, where $\Gamma$ and
$G$ are finite Abelian $p$-groups for some
prime number $p$. Assume that the finite local ring
$R \subseteq {\rm End}(G)$ defined in Lemma 3 is
principal. Then, the following assertions are equivalent.
\begin{enumerate}
\item[\rm (1)] $(\Gamma, G, \eta)$ is a minimal 
non-Kneser structure.
\item[\rm (2)] One of the following conditions is
satisfied:
\begin{enumerate}
\item[\rm (i)] $n = 1$, \emph{i.e.}\ the characteristic
of $R$ is $p$, and $R \cong \mathbb{F}_p[x]/(x^m)$
with $m \geq 2$ and $(m, p) = 1$.
\item[\rm (ii)] $p = 2, n = 2$, \emph{i.e.}\ the characteristic
of $R$ is $4$, and $R \cong \mathbb{Z}/4\mathbb{Z} [x]/(f(x), 2x^e)$,
where $f \in \mathbb{Z}/4\mathbb{Z}[x]$ is an Eisenstein polynomial
of degree $e \geq 1$, in particular, the nilpotency
index $m = 2 e$ is even and $R$ is free of rank
$e$ as $\mathbb{Z}/4\mathbb{Z}$-module.
\item[\rm (iii)] $p = 2, n = 2$, \emph{i.e.}\ the characteristic
of $R$ is $4$, and $R \cong \mathbb{Z}/4\mathbb{Z} [x]/(f(x), 2x^t)$,
where $f \in \mathbb{Z}/4\mathbb{Z} [x]$ is an Eisenstein polynomial
of degree $e \geq 2, 0 < t < e$, and the nilpotency
index $m = e + t$ is odd.
\end{enumerate}
\end{enumerate}
In all three cases above, the $R$-module $G$ can
be identified with $R^+, R/\mathfrak{m} \cong \mathbb{F}_p,
\Gamma = 1 + \mathfrak{m}$, and the injective cocycle 
$\eta \in Z^1(\Gamma, R^+)$ is unique up
to multiplication with elements from $R^\ast$ and
summation with homomorphisms from $\Gamma$ to 
$H^0(\Gamma, R^+) \cong \mathbb{Z}/p\mathbb{Z}$.
\end{proposition}

{The next result is an immediate consequence of
Propositions 4 and 5.}

\begin{corollary} \emph{\cite[Corollary 7.11]{Basarab4}}
Let $(\Gamma, G, \eta)$ be a minimal non-Kneser
structure. Assume that $G$ is Abelian of exponent
$e$ and the action of $\Gamma$ is induced by a character
$\chi : \Gamma \longrightarrow (\mathbb{Z}/e\mathbb{Z})^\ast$, \emph{i.e.}\
$\gamma g = \chi(\gamma) g$ for all $\,\gamma \in \Gamma, g \in G$.
Then $G \cong \mathbb{Z}/e\mathbb{Z}$, where $e$ is either an odd
prime number $p$ or $4$. In the first case, 
$(\Gamma, G, \eta) \cong (U, \mathbb{F}_p^+, u \mapsto u - 1)$,
where $U \cong \Gamma \cong \mathbb{Z}/r\mathbb{Z},\,2 \leq r\,|\,(p - 1)$,
is the unique subgroup of order $r =\,|\Gamma|$ of the
multiplicative group $\,\mathbb{F}_p^\ast$ of the prime field
$\,\mathbb{F}_p$, acting by multiplication.
In the latter case, $(\Gamma, G, \eta) \cong (\mathbb{Z}/2\mathbb{Z}, 
\mathbb{Z}/4\mathbb{Z}, 1\,{\rm mod}\,2 \mapsto 1\,{\rm mod}\,4)$ with
the unique nontrivial action of 
$\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{Z}/4\mathbb{Z}$.
\end{corollary}

{As a consequence of Corollary 6, we find again 
\cite[Lemma 1.18, Theorem 1.20]{Basarab1}}.

\subsection*{From quasi-co-Galois subgroups to quasi-co-Galois
field extensions}

{Let $\Gamma$ be a profinite group acting continuously on
a discrete quasicyclic group $A \subseteq \mathbb{Q}/\mathbb{Z}$.
Put $\mathfrak{Z} := Z^1(\Gamma, A)^\vee = {\rm Hom}(Z^1(\Gamma, A), A)$.
The Abelian profinite group $\mathfrak{Z}$ becomes a 
compact $\Gamma$-module with the action defined by
$(\gamma \psi)(g) = \gamma (\psi(g))$ for $\gamma \in \Gamma,
\psi \in \mathfrak{Z}, g \in Z^1(\Gamma, A)$, while the
continuous map $\eta : \Gamma \longrightarrow \mathfrak{Z},
\gamma \mapsto \eta(\gamma) : g \mapsto g(\gamma)$ is a
generating cocycle. We denote by $\mathbb{L}(G)$ the
lattice of all subgroups of a subgroup 
$G \subseteq Z^1(\Gamma, A)$ and by 
$\mathbb{L}(\Gamma\,|\,\Delta)$ the lattice of all
closed subgroups of $\Gamma$ lying over a closed
subgroup $\Delta \subseteq \Gamma$. The lattices 
$\mathbb{L}(Z^1(\Gamma, A))$ and $\mathbb{L}(\Gamma) :=
\mathbb{L}(\Gamma\,|\,\{\,1\,\})$ are related through
the canonical reversing maps $G \mapsto G^{\perp} :=
\displaystyle\bigcap_{g \in G} {\rm Ker}(g), \Delta
\mapsto \Delta^{\perp} := \{\,g \in Z^1(\Gamma, A)\,|\,
\Delta \subseteq {\rm Ker}(g) := g^{\perp}\,\}$.} 

{First we recall some basic notions of the cyclotomic 
abstract co-Galois theory.}

\begin{definition}
A subgroup $G \subseteq Z^1(\Gamma, A)$ is \emph{Kneser}
if the induced cocycle $\eta_G \in Z^1(\Gamma, \mathfrak{Z}/G^\vee)$
is surjective, so the induced map $\Gamma/G^{\perp} 
\longrightarrow \mathfrak{Z}/G^{\vee}$ is a homeomorphism
of profinite spaces.

$G$ is \emph{co-Galois} if $G$ is Kneser and the
lattices $\mathbb{L}(G)$ and
$\mathbb{L}(\Gamma\,|\,G^{\perp})$ are  canonically
anti-isomorphic.
 
A closed subgroup $\Delta \subset \Gamma$ is \emph{Kneser}
if there exists a Kneser subgroup $G \subseteq Z^1(\Gamma, A)$
such that $\Delta = G^{\perp} = {\rm Ker}(\eta_G)$.

$\Delta$ is \emph{hereditarily Kneser} if each
$\Lambda \in \mathbb{L}(\Gamma\,|\,\Delta)$ is Kneser.

$\Delta$ is \emph{co-Galois} if there exists a 
co-Galois subgroup $G \subseteq Z^1(\Gamma, A)$
such that $\Delta = G^\perp$. [Such a co-Galois
subgroup $G$ is unique].

$\Delta$ is \emph{quasi-co-Galois} if $\Delta$ is
maximal with the property that it is hereditarily
Kneser but not co-Galois.
\end{definition}

The quasi-co-Galois subgroups of $\Gamma$ are 
exactly the open Kneser subgroups $\Delta$
which are not co-Galois, but each $\Lambda \in
\mathbb{L}(\Gamma\,|\,\Delta) \setminus \{\,\Delta\,\}$
is co-Galois. A more precise description of the
quasi-co-Galois subgroups is given by the next
statement.

\begin{proposition} \emph{\cite[Lemma 3.1]{Basarab5}} 
The quasi-co-Galois subgroups of $\Gamma$ are exactly
those open subgroups $\Delta \subseteq \Gamma$ which satisfy
one of the following four conditions.
\begin{enumerate}
\item[\rm (1)] $\Delta \vartriangleleft \Gamma$,
$\widehat{1/4} \in A^{\Delta} \setminus A^{\Gamma}$, and
$\,\Gamma/\Delta \cong (\epsilon_4^{\perp}/\Delta) 
\rtimes (\Gamma/\epsilon_4^{\perp}) \cong \mathbb{D}_8$.
\item[\rm (2)] $\Delta \vartriangleleft \Gamma,\,\widehat{1/8} \in
A^{\Delta}, \,\widehat{1/4} \not \in A^{\Gamma}$, and the factor
group $\,\Gamma/\Delta \cong Q$ \emph{(the group of quaternions)}
has the presentation
\begin{multline*}\Gamma/\Delta \cong \langle\,\sigma, \tau\,|\,\sigma^4 = 1, 
\sigma^2 = \tau^2, \sigma \tau \sigma^{- 1} = \tau^{- 1}\,
\rangle =\\ \langle\,\sigma, \tau\, |\,\sigma \tau = 
\tau^{- 1} \sigma,\,\tau \sigma = \sigma^{- 1} \tau \,
\rangle,
\end{multline*}
with the action on $\,(1/8)\mathbb{Z}/\mathbb{Z} 
\subseteq A^{\Delta}\,$ given by
$\,\sigma\,\widehat{1/8} = \widehat{3/8}\,$ and 
$\,\tau\,\widehat{1/8} = \widehat{1/8}$.
\item[\rm (3)] $\widehat{1/8} \in A \setminus A^{\Delta},
\widehat{1/4} \in A^{\Delta} \setminus A^{\Gamma},
\Delta \cap \epsilon_8^{\perp} \vartriangleleft \Gamma$,
and the factor group $\Gamma/(\Delta \cap \epsilon_8^{\perp})$
has the presentation
$$\Gamma/(\Delta \cap \epsilon_8^{\perp})\cong 
(\epsilon_4^{\perp}/(\Delta \cap \epsilon_8^{\perp})) 
\rtimes (\Gamma/\epsilon_4^{\perp}) \cong$$
$$\langle\,\sigma, \tau, \delta\, |\, \sigma^2 = 
\tau^4 = \delta^2 = (\sigma \tau)^2 = 
[\sigma, \tau \delta] = [\tau,\delta] = 1\,\rangle ,$$
with the action on $\,(1/8)\mathbb{Z}/\mathbb{Z}\,$ given by
$\,\sigma\, \widehat{1/8} = - \widehat{1/8}, 
\tau \widehat{1/8} = \widehat{1/8}\,$
and $\,\delta \,\widehat{1/8} = \widehat{5/8}$.
\item[\rm (4)] $\Delta \vartriangleleft \Gamma$, there exist an
odd prime number $p$ and a divisor  $r>1$
of  $p - 1$ such that $\widehat{1/pr} \in A^{\Delta}$,
$\widehat{1/p} \not \in A^{\Gamma},\,\widehat{1/l} \in
A^{\Gamma}$ for all $l \in \cal{P}$ with $l\,|\, r$, and
$$\Gamma/\Delta\cong (\epsilon_p^{\perp}/\Delta)
\rtimes (\Gamma/\epsilon_p^{\perp}),\,
\Gamma/\epsilon_p^{\perp}\cong\mathbb{Z}/r\mathbb{Z},
\,\epsilon_p^{\perp}/\Delta\cong\mathbb{Z}/p\mathbb{Z}.$$
\end{enumerate}
\end{proposition}

{[For $n \in \mathbb{N},\,\epsilon_n$ denotes the
coboundary $\gamma \mapsto (\gamma - 1) \widehat{1/n}$,
while $\cal{P}$ stands for the set of all prime natural
numbers where $2$ is replaced by $4$.]}

{The field theoretic notions which are required for stating
the field theoretic analogue of Proposition 7 above are briefly
recalled in the following.}

{Let $E/F$ be an algebraic field extension. We
denote by $T(E/F)$ the multiplicative group of all 
elements $x \in E^\ast$ which are radicals over $F$, \emph{i.e.}\ 
$x^n \in F^\ast$ for some $n \in \mathbb{N} \setminus \{\,0\,\}$.
The \emph{co-Galois group} ${\rm Cog}(E/F)$ of the 
field extension $E/F$ is the factor group $T(E/F)/F^\ast$, 
\emph{i.e.}\ the torsion subgroup of the factor group $E^\ast/F^\ast$.
By Hilbert's Satz 90, ${\rm Cog}(E/F)$ is canonically
isomorphic with $Z^1(\Gamma, \mu)$ whenever $E/F$ is
Galois with Galois group $\Gamma$ and $\mu$ is the multiplicative
quasicyclic group of all roots of unity contained in $E$. 
A dictionary relating the group theoretic notions and
the the corresponding field theoretic ones is transparent.}

\begin{definition}
A finite separable field extension $E/F$ is \emph{Kneser} if
$E = F(G)$ for some subgroup $G \subseteq T(E/F)$ lying
over $F^\ast$ such that $[E : F] = (G : F^\ast)$. If,
in addition, the lattice $\mathbb{L}(E/F)$ of all 
intermediate fields between $F$ and $E$ and the lattice
$\mathbb{L}(G\,|\,F^\ast)$ are canonically isomorphic, we
say that $E/F$ is \emph{co-Galois}. [In the latter case,
the group $G$ of radical elements is uniquely determined.]

An algebraic separable field extension $E/F$ is 
\emph{hereditarily Kneser} if each finite subextension
$L/F$ is Kneser. $E/F$ is \emph{co-Galois} if each
finite subextension $L/F$ is co-Galois.

An algebraic separable field extension $E/F$ is called
\emph{quasi-co-Galois} if it is minimal with the property
that it is hereditarily Kneser but not co-Galois.
\end{definition}

It turns out that the quasi-co-Galois field extensions
are exactly those separable finite field extensions $E/F$
which are Kneser, not co-Galois, but each subextension
$L/F, L \neq E$ is co-Galois. As a consequence of
Proposition 7, we obtain a precise description of the
quasi-co-Galois field extensions as follows.

\begin{proposition} \emph{\cite{Basarab5}}
Let $E/F$ be an algebraic separable field extension. Then,
the necessary and sufficient condition for $E/F$ to be
quasi-co-Galois is that one of the following situations hold :
\begin{enumerate}
\item[\rm (1)] {\rm char}$(F) \neq 2, \zeta_4 \not \in F$,
and $E = F(\zeta_4, \theta)$ for some $\theta \in E$
satisfying $\theta^4 = a$, where $a \in F \setminus \pm F^2$.
\item[\rm (2)] {\rm char}$(F) \neq 2, 2 \in (- F^2) \setminus
F^2$, and $E = F(\theta)$ for some $\theta \in E$ such that
$a := \theta^8 \in (- F^2) \setminus (- F^4)$.
\item[\rm (3)] {\rm char}$(F) \neq 2, 2 \not \in \pm F^2,
\zeta_4 \not \in F$, and $E = F(\theta)$ for some $\theta \in E$
such that $a := \theta^8 \in (- F^2) \setminus (- F^4 \cup
- 4 F^4)$.
\item[\rm (4)] there exist a prime number $p \neq 2$ and
$1 < r\; |\; (p - 1)$ such that the characteristic
exponent of $F$ is prime to $pr, \zeta_p \in 
E \setminus F, \zeta_r \in F(\zeta_p), \zeta_l \in F$ 
for all $l \in {\cal{P}},\, l\,|\,r, [F(\zeta_p) : F] = r$, 
and $E = F(\zeta_p, \theta)$ for some $\theta \in E$ 
satisfying $\theta^p = a$, where $0 \neq a \in F \setminus F^p$.
\end{enumerate}
[For $n \in \mathbb{N} \setminus \{\,0\,\},\,\zeta_n$
denotes a primitive $n$th root of unity.]
\end{proposition}

\subsection*{From strongly co-Galois subgroups to strongly
co-Galois field extensions}

\begin{proposition} \emph{\cite[Lemma 2.8]{Basarab5}}
Let $\Delta$ be a \emph{radical subgroup} of $\Gamma$,
\emph{i.e.}\ $\Delta = \Delta^{\perp \perp}$. Then the following
statements are equivalent 
\begin{enumerate}
\item[\rm (1)] $\Delta^{\perp}$ is a co-Galois subgroup 
of $Z^1(\Gamma, A)$.
\item[\rm (2)] $\Delta^{\perp}$ is a Kneser subgroup of 
$Z^1(\Gamma, A)$.
\item[\rm (3)] $A^\Gamma[p] = A^\Delta[p]$ for all
$p \in \cal{P}$.
\end{enumerate}  
\end{proposition}

{Call such a subgroup} $\Delta \subseteq \Gamma$ 
{strongly co-Galois}

{The field theoretic analogue of Proposition 9
due to Greither and Harrison \cite{Basarab7} reads as follows.}

\begin{theorem}
The following statements are equivalent for a separable
radical extension $E/F$.
\begin{enumerate}
\item[\rm (1)] $E/F$ is $T(E/F)$-Kneser.
\item[\rm (2)] $E/F$ is $T(E/F)$-co-Galois.
\item[\rm (3)] $\mu_p(E) \subset F$ for all $p \in \cal{P}$.
\end{enumerate}
\end{theorem}
{Call such an extension} $E/F$ strongly co-Galois.

{The following class of strongly co-Galois extensions 
plays a key role in the model theory of Henselian fields
and $p$-adically closed fields.}

\begin{theorem} \emph{\cite[Theorem 3.8]{Basarab8}, \cite[Proposition 1.1]
{Basarab2}}
Let $(F, v)$ be a Henselian valued field of characteristic 
zero. An algebraic field extension $E$ of $F$ is 
strongly co-Galois whenever $E$ is a core-dense extension of $F$.

In particular, if $F$ is a $p$-valued Henselian field,
$E/F$ is strongly co-Galois whenever $E$ is of
the same $p$-rank as $F$.
 \end{theorem}

{Note that the radical structure theorem above is a
main ingredient in the proof of an isomorphism theorem
for algebraic extensions \cite[Corollary 3.11]{Basarab8},
\cite{Basarab6}, and of a (relative) quantifier elimination
theorem \cite[Theorem 5.6]{Basarab8}, \cite[Theorems A, B]
{Basarab2}}.

\begin{ourbiblio}{99} 
		       
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\bibitem{Basarab7}
C. Greither and D.K. Harrison, \emph{A Galois correspondence
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\presenteraddress{Institute of Mathematics ``Simion Stoilow''
of the Romanian Academy}

\email{serban.basarab@imar.ro}

\web{www.imar.ro/~sbasarab/} 




