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\noindent
\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Mar. $\mathbf{23}^\mathrm{\bf rd}$, 2016}

\vspace{0.2in}

\begin{description}
\item[Irreducible Problems] Let $\mathbb{F}$ be a field.
   \begin{enumerate}[(a)]
   \item Let $f(x)= a_nx^n+\cdots+a_1x+a_0 \in \mathbb{F}[x]$ be irreducible. Prove that $g(x) = a_0x^n+\cdots+a_{n-1}x+a_n$ (reverse the order of the coefficients) is irreducible in $\mathbb{F}[x]$ as well.\\
    {\it Note/Hint:} $x^nf(1/x)=g(x)$ is called the {\it reciprocal polynomial} of $f(x)$.
   \item Use part (a) and Eisenstein's criterion to show $h(x)=12x^5-24x^4+6x^2+18x-1$ is irreducible in $\mathbb{Q}[x]$.
   \item Show $\ell(x)=5x^3+9x^2-2x+2$ is irreducible in $\mathbb{Q}[x]$ by reducing modulo $p$ for some prime $p$.
   \item We could also show that $\ell(x)$ is irreducible using the rational root theorem. Sketch out such a proof.\\ 
             {}[You don't actually have go through the trouble of plugging numbers into $\ell(x)$.]
   \end{enumerate}

\item[Page 49 \#71] Find the roots of the polynomials $f(x) \in \mathbb{Z}[x]$ using the cubic / quartic formula.
   \begin{description}
   \item[{\normalfont (iii)}] $f(x) = x^3-24x^2-24x-25$
   \item[{\normalfont (vi)} (Grad Problem)] $f(x) = x^4-15x^2-20x-6$
   \end{description}
   
\item[Page 58 \#72] Let $\mathbb{E}/\mathbb{F}$ be an extension (of fields). 
   \begin{enumerate}[(a)]
   \item Let $\alpha,\beta \in \mathbb{E}$ be algebraic over $\mathbb{F}$ and $\alpha \not=0$. Prove that $\alpha+\beta$, $\alpha\beta$, and $\alpha^{-1}$ are also algebraic over $\mathbb{F}$. \\
{}[{\it Hint:} Don't try to find polynomials. Instead, use the degree formula prove $\mathbb{F}(\alpha,\beta)$ is finite dimensional over $\mathbb{F}$.]
   \item Let $\mathbb{K} = \{ \alpha \in \mathbb{E} \;|\; \alpha \mbox{ is algebraic over } \mathbb{F} \}$. Prove that $\mathbb{K}$ is a subfield of $\mathbb{E}$ containing $\mathbb{F}$.
   \item {}{\bf (Grad Problem)} Define $\mathbb{A} = \{ \alpha \in \mathbb{C} \;|\; \alpha \mbox{ is algebraic (over } \mathbb{Q}\mbox{)}\}$. This is called the field of {\it algebraic numbers}. Briefly explain why $\mathbb{A}/\mathbb{Q}$ is an algebraic extension. Then prove it is {\bf not} a finite extension.
   \end{enumerate}

\end{description}

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