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\begin{document}

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


\begin{center}
{\bf PROBLEM SET 7 -- Due April 9, 2026}
\end{center}


\skipline
\noin 
{\bf A1)} Let $n\geq 1$ be positive integer and
\[
\zeta_n:=e^{2\pi i/n}=\cos(\frac{2\pi}{n})+i\sin(\frac{2\pi}{n}).
\]
Find the minimal polynomial over $\Q$ of (a) $\zeta_5$ \  (b) $\zeta_6$
\ (c) $\zeta_8$ \ (d) $\zeta_{10}$.

\skipline \noin {\bf A2)} Let $K_1$ and $K_2$ be two finite extensions
of a field $F$ which are contained in the field $L$. Assume that
both $K_1$ and $K_2$ are splitting fields (of some polynomials) over
$F$. Prove that the composite field $K_1K_2$ and the intersection
$K_1\cap K_2$ are both splitting fields over $F$.



\skipline \noin {\bf A3)} Let $K_1$ and $K_2$ be two finite extensions
of a field $F$ which are both contained in the field $L$. Prove that
the $F$-algebra $K_1\otimes_F K_2$ is a field if and only if
$[K_1K_2:F]=[K_1:F][K_2:F]$.


\skipline
\noin
{\bf A4)} Show that if $F$ is an infinite field, then the multiplicative group
$(F^*,\cdot)$ is not cyclic.


\skipline \noin {\bf A5)} (a) Let $p$ be a prime and $f(x)$ be an
irreducible polynomial of degree $n$ over the finite field
$\F_p$. Show that $\F_p[x]/(f(x))$ is a field with $p^n$ elements,
which is isomorphic to a splitting field of the polynomial $x^{p^n}-x$
and $f(x)$ divides $x^{p^n}-x$.


\medskip
\noin
(b) Let $g(x)$ be an irreducible polynomial in $\F_p[x]$. Show that $g(x)$ divides $x^{p^n}-x$ if and only if $\deg(g(x))$ divides $n$.

\skipline
\noin
{\bf A6)} (a) Prove that $x^3+x^2+1$ and $x^3+x+1$ are irreducible over
the finite field $\F_2$.

\medskip
\noin
(b) Give an explicit isomorphism between the quotients $\F_2[x]/(x^3+x^2+1)$ and
$\F_2[x]/(x^3+x+1)$. 


\skipline
\noin
 {\bf A7)} For any prime $p$ and non-zero $a\in \F_p$, prove that
 the polynomial  $x^p-x+a$ is irreducible and separable over $\F_p$.
 [Hint: Prove that if $\al$ is root then $\al+1$ is also a root.]

 
\skipline
\noin 
{\bf B1)}  A complex number $z$ is called an {\em
  algebraic integer} if $z$ is a root of a monic polynomial
$f(x)=x^n+a_{n-1}x^{n-1}+\cdots + a_1 x + a_0$ with integer coefficients $a_i$,
for some $n\geq 1$.

\medskip
\noin
(a) Prove that if $\al_1,\ldots,\al_k$ are algebraic integers, then
$\Z[\al_1,\ldots,\al_k]$ is a finitely generated abelian group.

\medskip
\noin
(b) Prove that the set $\mathcal{O}$ of all algebraic integers is a
{\em subring} of $\C$.

\medskip
\noin
(c) Suppose that $z$ is a root of a monic polynomial
$g(x)=x^n+\al_{n-1}x^{n-1}+\cdots + \al_1 x + \al_0$ with
coefficients $\al_i\in \mathcal{O}$. Prove that
$z\in \mathcal{O}$.


\skipline
\noin 
{\bf B2)} Let $F$ be any field.

\medskip
\noin
(a) Suppose that the additive group $(F,+)$ is a finitely generated 
abelian group. Prove that $F$ must be a finite field.

\medskip
\noin
(b) Suppose that $F$ is a finite field. Classify the additive subgroup
$(F,+)$ up to group isomorphism. 


\skipline
\noin 
{\bf B3)} Let $F$ be a field and $F(x)$ be the quotient field of the
polynomial ring $F[x]$. Let $t\in F(x)$ be the rational function
$t=p(x)/q(x)$, where $p(x),q(x)\in F[x]$ are relatively prime polynomials
and $q(x)\neq 0$. Then $F(x)$ is an extension of the field $F(t)$.

\medskip
\noin
(a) Show that the polynomial $p(z)-tq(z)$ in the variable $z$ 
and coefficients in $F(t)$ is irreducible over $F(t)$ and
has $x$ as a root. [Hint: By Gauss' Lemma this polynomial is
irreducible in $(F(t))[z]$ if and only if it is irreducible in
$(F[t])[z]$. Then note that $(F[t])[z]=(F[z])[t]$.]

\medskip
\noin
(b) Show that $[F(x):F(t)] = \max(\deg p(x), \deg q(x))$.


\medskip
\noin
(c) Show that $F(t)=F(x)$ if and only if $$t(x)=\frac{ax+b}{cx+d}$$ 
for some $a,b,c,d\in F$ such that $ad-bc \neq 0$.




\skipline


\begin{center}
{\bf \underline{C problem}}
\end{center}

\skipline
\noin
{\bf C1)} Let $\phi$ denote the Frobenius map $x\mapsto x^p$ on the 
finite field $\F_{p^n}$. Consider $\phi$ as an $\F_p$-linear transformation
of the $n$-dimensional $\F_p$-vector space $\F_{p^n}$. 

\medskip
\noin
(a) Determine the rational canonical form over $\F_p$ for $\phi$.

\medskip
\noin
(b) Determine the Jordan canonical form for $\phi$ (over a field 
which contains all the eigenvalues of $\phi$).






\end{document}
