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\begin{document}

\begin{center}
{\bf Math 601 -- Spring 2026 -- Harry Tamvakis}
\end{center}


\begin{center}
{\bf PROBLEM SET 9 -- Due April 23, 2026}
\end{center}


    

\skipline
\noin
{\bf A1)} Let $K/F$ be a Galois extension whose Galois
group is the symmetric group $S_3$. Is it true that $K$ is the
splitting field of an irreducible cubic polynomial over $F$? [Hint:
Find an intermediate subfield $E=F(\alpha)$ of degree 3 over $F$ and
consider the minimal polynomial of $\alpha$.]

\skipline
\noin
{\bf A2)} Let $L/F$ be a Galois extension of degree $15$.

\medskip
\noin
(a) Show that every subextension $K$ with $F\subset K \subset L$ 
is normal.

\medskip
\noin
(b) Suppose that $f(x)\in F[x]$ is irreducible and its splitting field is $L$.
Show that $\deg(f(x))=15$.


\skipline
\noin
{\bf A3)} Let $F$ be an algebraically closed field.
Show that there is an infinite subfield of
$F$ that is algebraic over the prime subfield of $F$, but
is not algebraically closed.


\skipline
\noin {\bf A4)} Suppose $K/\Q$ is a Galois extension such that
$\text{Gal}(K/\Q) = A_5$, the group of
even permutations of 5 letters. Show that there is an irreducible polynomial
$f(x)\in\Q[x]$ of degree $20$ such that $K$ is the splitting field of $f(x)$. 


\skipline
\noin
{\bf A5)} Let $\alpha$ be algebraic over $\Q$ and let $L$ be 
the Galois closure of $\Q(\alpha)/\Q$, that is, the smallest Galois
extension of $\Q$ which contains $\Q(\alpha)$. Let $p$ be a prime
dividing the order of $G(L/\Q)$. Prove that there exists a subfield
$K$ of $L$ with $[L:K]=p$ and $L=K(\alpha)$. 


\skipline
\noin {\bf B1)} Let $f\in \Q[x]$ be irreducible of degree $p$, where $p$ is 
a prime. Let $K$ be the splitting field of $f$ and suppose that there 
are roots $\alpha$ and $\beta$ of $f$ such that $K=\Q(\alpha,\beta)$. 
We regard the Galois group $G=G(K/\Q)$ as a subgroup of the symmetric
group $S_p$.

\medskip
\noin
(a) Show that $|G|\leq p(p-1)$. 

\medskip
\noin
(b) Show that $G$ contains a $p$-cycle $\sigma$.

\medskip
\noin
(c) Prove that the subgroup $H=\langle\sigma\rangle$ of $G$ generated by 
$\sigma$ is a normal subgroup of $G$.

\medskip
\noin
(d) Prove that $G/H$ is a cyclic group of order dividing
$p-1$. Deduce that $G$ is solvable.

\medskip
\noin
{\bf Remark:} Galois proved the converse statement: If $G$ is solvable
for an irreducible polynomial $f$ of prime degree, then the splitting 
field can be generated by two roots of $f$.


\skipline
\noin
{\bf B2)} Let $F$ be a subfield of the real numbers $\R$. Suppose that
$a\in F$, let $\root{n}\of{a}$ be a real $n$-th root of $a$, and set
$L=F(\root{n}\of{a})$. Prove that if $K$ is any Galois extension 
of $F$ with $K\subset L$ then $[K:F]\leq 2$. 


\skipline
\noin
{\bf B3)} Let $K\supset F$ be a finite extension of fields
with Galois group $G=G(K/F)$. Then $G$
acts on the elements of $K$. By a {\em Galois basis} of $K$ over
$F$ we mean a linear basis of $K$ over $F$ that is the orbit
under $G$ of a single element of $K$.

\medskip
\noin
(a) Show that if a Galois basis of $K$ over $F$ exists, then
$K$ is a Galois extension of $F$.


\medskip
\noin
For the next three questions assume that $K\supset F$ is Galois
and $G=\{\sigma_1,\ldots,\sigma_n\}$.

\medskip
\noin
(b) Given $\alpha\in K$, show that
the set $\{\sigma(\alpha)\ |\ \sigma\in G\}$ is a
Galois basis of $K$ over $F$ if and only if the matrix
\[
\left\{\sigma_i\sigma_j(\alpha)\right\}_{1\leq i,j \leq n}
\]
has non-zero determinant.

\medskip
\noin
(c) If $f\in F[x_1,\ldots,x_n]$ and
\[
\dis
f(\sigma_1(\alpha),\ldots,\sigma_n(\alpha))=0
\]
for every $\alpha\in K$, prove that $f=0$. [Hint: For a basis
$\{b_i\}$ of $K$ over $F$ set
\[
\dis
g(y_1,\ldots,y_n):=f\left(\sum_iy_i\sigma_1(b_i),\ldots,\sum_iy_i
\sigma_n(b_i)\right),
\]
and deduce that $g=0$.]

\medskip
\noin
(d) For each $i$ define a variable $x_{\sigma_i}$ indexed by
$\sigma_i$ and set
\[
f(x_{\sigma_1},\ldots,x_{\sigma_n}):=\det(x_{\sigma_i\sigma_j}).
\]
Show that $f(1,0,\ldots,0) \neq 0$ and deduce that a Galois basis
of $K$ over $F$ always exists.





\end{document}
