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Unverified Commit 9fcdce19 authored by rrueger's avatar rrueger
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Algebra II: Lecture 11. Complete section on characteristic p

parent ae05ec93
......@@ -109,16 +109,19 @@ which situations such an $\hat{E}$ exists.
In the case of $F$ having characteristic $p \neq 0$, there is no such $\hat E$
in general. To show this, we pose the following counterexample:
Let $F = \F _p(X)$. Note, this field is infinite. Moreover, let $E = F(r)$ for
$r$ a root of $p(Z) = Z^p - X$. We immediately now notice that $p'(Z) = 0$, so
our previous line of argument doesn't work.
Since $r$ is a root, $0 = p(r) = r^p - X$ implying $X = r^p$ so $P(Z) = Z^p -
r^p$, and as we are in characteristic $p$ we have that $p(Z) = Z^p - r^p = {(Z
- r)}^p$. This means that $E$ is the splitting field over a polynomial with
repeated roots. We know that the automorphism group of $E/F$ can only permute
the roots, and as there is only one root, $\Aut (E/F)$ is trivial. The extension
is not Galois.
Let $F = \F _p(X)$. Moreover, let $E = F(r)$ for $r$ a root of $p(Z) = Z^p - X$.
We immediately now notice that $p'(Z) = 0$, so our previous line of argument
doesn't work. Since $r$ is a root, $0 = p(r) = r^p - X$ implying $X = r^p$ so
$P(Z) = Z^p - r^p$; as we are in characteristic $p$ we have that $p(Z) = Z^p -
r^p = {(Z - r)}^p$. This means that $E$ is the splitting field over a polynomial
with a single repeated root. We know that the automorphism group of $E/F$ can
only permute the roots, and as there is only one root, $\Aut (E/F)$ is trivial.
The extension is not Galois. The question is now, \textit{is there are larger
$\hat E \supseteq E$ such that $\hat E /F$ is Galois?} Assume there was such an
$\hat E$, then for any automorphism $\sigma \in \Aut (\hat E /F)$ it holds that
$X \in F = \F _p (X)$ is fixed, so is $X^p$ and thus also $r$. It follows that
$F(r) = E$ is also fixed by $\sigma$ and thus not \textbf{only} $F$ is fixed,
meaning the extension is not Galois by the very first definition of a Galois
extension.
\section{Skew Fields and Wedderburn's Theorem}
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