### 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}
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!