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\begin{document}

\begin{center}
{\bf Math 100C: Spring 2017 \\ Homework 4 \\ Due 5/10/2017}
\end{center}





\vspace{0.1in}
\noindent
{\bf Problem 1:}  
Let $F$ be a field and let $R$ be an integral domain with $F \subseteq R$.  Suppose that 
$R$ is finite-dimensional as an $F$-vector space.  Prove that $R$ is a field.
(This is not true without the finite-dimensionality assumption; consider $F \subseteq F[x]$.)

\vspace{0.1in}
\noindent
{\bf Problem 2:}  
Let $F$ be a field and let $f(x) \in F[x]$ be a polynomial of degree $d > 0$.  Prove that the set
$$
\BBB := \{ \overline{1}, \overline{x}, \dots, \overline{x}^{d-1} \}
$$
is a basis for the quotient $F[x]/(f(x))$, considered as an $F$-vector space.
\footnote{Here $\overline{g(x)} = g(x) + (f(x)) \in F[x]/(f(x))$ is my notation for cosets.
I referenced this fact in class, but I heard that you hadn't seen it before.  Sorry about that.}

\vspace{0.1in}
\noindent
{\bf Problem 3:} 
Let $K/F$ be an extension of fields and let $\alpha \in K$ be algebraic over $F$.  Suppose the 
irreducible polynomial for $\alpha$ over $F$ is
$$
f(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0,
$$
with the $c_i \in F$.  Find an explicit formula for $\alpha^{-1}$ in terms of $\alpha$ and the $c_i$.



\vspace{0.1in}
\noindent
{\bf Problem 4:}  
Let $K/F$ be a field extension and suppose that $\alpha \in K$ has degree $5$ over $F$.  Prove
that $F(\alpha) = F(\alpha^2)$.


\vspace{0.1in}
\noindent
{\bf Problem 5:}
Let $\zeta_n = \exp(2 \pi i/n)$.  Prove that $\zeta_5 \notin \QQ(\zeta_7)$.

\vspace{0.1in}
\noindent
{\bf Problem 6:}
Let $F \subseteq L \subseteq K$ be a tower of field extensions.  Suppose that both of the 
intermediate extensions $F \subseteq L$ and $L \subseteq K$ are algebraic.  Prove that the big
extension $F \subseteq K$ is algebraic.
\footnote{An extension of fields $F \subseteq K$ is {\em algebraic} if every element of $K$ is algebraic
over $F$.}

\vspace{0.1in}
\noindent
{\bf Problem 7:}
Let $\alpha = \sqrt{3} + \sqrt{5}$.  Find the irreducible polynomial for $\alpha$ over $\QQ$.
What about over $\QQ(\sqrt{5})$?  What about over $\QQ(\sqrt{15})$?

\vspace{0.1in}
\noindent
{\bf Problem 8:} 
(Optional - not to be handed in.)
We continue our study of the representation theory of the circle group $S^1$.  Let 
$R: S^1 \rightarrow GL_n(\CC)$ be an irreducible continuous $n$-dimensional complex
 representation of $S^1$.  We will show that $n = 1$.
 \begin{enumerate}
 \item  Let $z \in S^1$.  Prove that the matrix $R_z \in GL_n(\CC)$ has an eigenvector $v \in \CC^n - \{0\}$.
 \item  Prove that if $v$ is an eigenvector for $R_z$, then $v$ is also an eigenvector for $R_{z^r}$ for
 $r \geq 1$.
 \item  For any $k = 1, 2, 3, \dots$ we consider the `dyadic' points on the circle $S^1$: 
 $$
 A_k = \{ \exp(2 \pi i r / 2^k) \,:\, r = 1, 2, \dots, 2^k \},
 $$
 so that $A_1 \subset A_2 \subset \cdots $; we let $A = \bigcup_{k \geq 1} A_k$.  Prove that if $z \in A$,
 the matrix $R_z \in GL_n(\CC)$ is diagonalizable.
 \item  For any fixed $k$, let 
 $$\CC^n = E^{(k)}_1 \oplus \cdots \oplus E^{(k)}_{m_k}$$ 
 be the decomposition
 of $\CC^n$ into simultaneous eigenspaces for the matrices $\{ R_z \,:\, z \in A_k\}$.  Prove that if
 $$\CC^n = E^{(k+1)}_1 \oplus \cdots \oplus E^{(k+1)}_{m_{k+1}}$$ 
 is the corresponding decomposition for $A_{k+1}$, then the spaces $E^{(k+1)}_i$ are subspaces of the 
 spaces $E^{(k)}_j$ (and hence {\em refine} these spaces).
 \item  Prove that there is a nonzero vector $v \in \CC^n - \{0\}$ which is a simultaneous eigenvector of
 {\em all} of the operators $\{R_z \,:\, z \in A \}$.
 \item  Use continuity to conclude that the $1$-dimensional space $V = \mathrm{span}(v)$ is a $S^1$-submodule
 of $\CC^n$, so that by irreducibility $\mathrm{span}(v) = \CC^n$ and $n = 1$.
 \end{enumerate} 


\vspace{0.1in}
\noindent
{\bf Problem 9:} (Optional - not to be handed in.)
{\bf ``Maschke on the circle."}
Let $S^1$ be the circle group and let $V$ be a finite-dimensional $S^1$-module over $\CC$ (with, of course,
the map $\rho: S^1 \rightarrow GL(V)$ continuous).
Let $W \subseteq V$ be a $S^1$-submodule and let $\pi: V \rightarrow W$ be a $\CC$-linear projection.
Define a new map $\pi': V \rightarrow V$ by
$$
\pi'(v) = \int_{z \in S^1}  z.\pi(z^{-1}.v)  dz.
$$
(What does the integral over $S^1$ mean?  Don't worry if you do not know.  Just assume whatever 
plausible properties about it that you want!  This is what motivated its definition, anyway.  We normalize
so that $\int_{S^1} 1 dz = 1$.)  Prove that $\pi'$ is a $S^1$-homomorphism and a projection onto $W$,
so that $U := \mathrm{Ker}(\pi')$ is a $S^1$-submodule of $V$ with $V = W \oplus U$.

Deduce that if $V$ is any finite-dimensional $S^1$-module over $\CC$, then we may write
$$
V = W_1 \oplus \cdots \oplus W_r,
$$
where the $W_1, \dots, W_r$ are irreducible $S^1$-modules (which are $1$-dimensional by the previous problem!).


\vspace{0.1in}
\noindent
{\bf Problem 10:} (Optional - not to be handed in.)
{\bf ``Character orthogonality on the circle."}
If $\chi, \psi: S^1 \rightarrow \CC$ are any two continuous complex-valued functions on the circle $S^1$,
define their `inner product' by
$$
\langle \chi, \psi \rangle = \int_{z \in S^1} \chi(z) \cdot \overline{\psi(z)} dz.
$$
We have seen that the irreducible characters (= characters of irreducible representations) of $S^1$
are $\{ \chi_n \,:\, n \in \ZZ \}$, where
$$
\chi_n(z) = z^n.
$$
Verify that we have {\em character orthogonality}:
$$
\langle \chi_n, \chi_m \rangle = \delta_{n,m}.
$$


\end{document}








