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\centerline{ \bigbf Problems, October 11, 1989 }
\bs
\centerline { \bigbf Math 401 -- Second Course in Abstract Algebra, I}
\ms
\centerline { \bigbf Prof. Daniel Grayson }
\ms
\centerline { 2:00 PM MWF, Section F1, 241 Altgeld Hall }
\bs

\problem 7

Consider a field $K$ of characteristic $p$ and an automorphism $\sigma$ of $K$
of order $p$.  Let $F = \{ x \in K \,|\, \sigma(x) = x \}$ be the fixed field
of $\sigma$.  Study the structure of the field extension $F \subseteq K$.

\problem 8

Consider a field extension $F \subseteq K$, let $G$ be the group of
automorphisms of $K$ as an $F$-algrebra, and let $R$ the ring of endomorphisms
of $K$ as an $F$-vector space.  Notice that $G$ is a subgroup of $R^\times$.
We interpret $R$ as a vector space over $K$ by defining $(\alpha f)(\beta) =
\alpha \cdot f(\beta)$ for $\alpha, \beta \in K$ and $f \in R$.  Prove that
$G$, regarded as a subset of $R$, is linearly independent over $K$.

\problem 9

Consider a field $K$ and a subgroup $G$ of $\Aut(K)$.  Let $F = K^G$ be the
fixed subfield $\{ x \in K \mid \forall \sigma \in G, \sigma x = x \}$.
Consider the map $f : K \to \prod_G K$ defined by $f(x) = \(\sigma x\)_{\sigma
\in G}$.  Interpret $K$ as a vector space over $F$ and $\prod_G K$ as a vector
space over $K$.  Show that $f$ transforms sets linearly independent over $F$
into sets linearly independent over $K$.


\end

