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\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Jan. $22^\mathrm{nd}$, 2016}

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These problems are drawn from Rotman's ``Galois Theory'' ($2^{\mathrm{nd}}$ edition).

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\begin{description}
\item[Page 12 \#5] Show that the intersection of any family of subrings is itself a subring. 

Given $S_\alpha$ is a subring of a ring $R$ for all $\alpha \in I$ ($I$ is some index set). Show $\bigcap\limits_{\alpha \in I} S_\alpha$ is a subring of $R$.

\item[Page 12 \#8] Let $R$ be a (commutative) ring (with 1) and let $f(x)=r_0+r_1x+\cdots+r_nx^n \in R[x]$. We can define the formal derivative\footnote{This is a totally formal notion of derivative. There is no concept of ``limit'' in a general ring $R$. Also, keep in mind that $2r_2x$ is not $2$ times $r_2x$ but instead it is the $2^{\mathrm{nd}}$ additive power of $r_2x$. In other words, $2r_2x = r_2x+r_2x$. This may not show up in your proof, but it is something you should think about as you write up your solution.} of $f(x)$ as follows: $f'(x) = \dfrac{d}{dx}\Big[f(x)\Big] =  r_1+2r_2x+\cdots+nr_nx^{n-1}$. 

Prove that the derivative is linear and obeys the product rule: $[f(x)+g(x)]' = f'(x) + g'(x)$, $[cf(x)]' = cf'(x)$, and $[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)$ for all $f(x),g(x) \in R[x]$ and $c \in R$.

Keep in mind that if $f(x) = \sum\limits_{i=0}^n r_ix^i$ and $g(x) = \sum\limits_{j=0}^m s_jx^j$, then $f(x)g(x) = \sum\limits_{\ell=0}^{m+n} \left(\sum\limits_{k=0}^\ell r_ks_{\ell-k}\right)x^\ell$.

\item[Page 16 \#13] Degrees of Difficulty \qquad $\partial(f(x)) = \mathrm{deg}(f(x))=$ the degree of $f(x)$.
   \begin{enumerate}[i.]
   \item Let $R$ be an integral domain, $f(x),g(x) \in R[x]$, and $f(x),g(x) \not=0$. Briefly explain why the leading coefficient of $f(x)g(x)$ is the product of the leading coefficients of $f(x)$ and $g(x)$. Then justify why $\partial(f(x)g(x)) = \partial(f(x)) + \partial(g(x))$. 
   \item Prove that if $R$ is an integral domain, then so is $R[x]$.
   \item Consider $R=\mathbb{Z}_4[x]$. Show that $(2x+1)^2=1$. What does this say about the formula in part i and the result of part ii?
   \item Show that $x$ can be factored: $x=f(x)g(x)$ in $\mathbb{Z}_4[x]$ in such a way that neither $f(x)$ nor $g(x)$ is constant.
   \end{enumerate}
   
\item[Page 16 \#16] Field or not a field. 
   \begin{enumerate}[i.]
   \item Let $\mathbb{F}$ be a field. Show that $(\mathbb{F}[x])^\times = \mathbb{F} - \{0\}$ (i.e. the units of $\mathbb{F}[x]$ are exactly the non-zero constant polynomials).
   \item Show that $\mathbb{Z}_2[x]$ is an infinite ring with exactly 1 unit.
   \item Give an example of a non-constant polynomial in $\mathbb{Z}_4[x]$ that is a unit.
   \end{enumerate}
   
\item[Page 17 \#19] Prove that the intersection of any family of subfields is itself a subfield. (Note that this intersection isn't the trivial ring since all of the subfields contain 1.) 

Let $\mathbb{E}_\alpha$ be a subfield of a field $\mathbb{F}$ for all $\alpha \in I$ ($I$ is some index set). Show $\bigcap\limits_{\alpha \in I} \mathbb{E}_\alpha$ is a subfield of $\mathbb{F}$.
\end{description}

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