\documentclass[11pt]{article}
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\usepackage{xcolor}
\usepackage{verbatim}
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\usepackage{hyperref}
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    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Algebra},
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% THEOREM ENVIRONMENTS
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\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
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\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\BF}{\mathbb{F}} % field
% \newcommand{\kring}{\mathbf{k}} % commutative ring k (e.g. field)
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
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\newcommand{\F}{\mathbb{F}} % random field
\newcommand{\Isom}{\mathrm{Isom}} % isometries group

\DeclareMathOperator{\coker}{coker} % cokernel
\DeclareMathOperator{\img}{im} % image
\DeclareMathOperator{\sgn}{sgn} % sign of permutation
\DeclareMathOperator{\ord}{ord} % order of group element
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% \DeclareMathOperator{\gcd}{gcd} % greatest common divisor
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\newcommand{\Mat}{\mathrm{Mat}} % matrices





% SOME ENVIRONMENTS
\newcommand{\exheading}[1]{\section*{Exercise #1}}
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% \setcounter{exnum}{1} % default 0 start
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\newcommand{\ee}{\end{enumerate}}
% round matrix
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\newcommand{\epmat}{\end{pmatrix}}
% square matrix
\newcommand{\bbmat}{\begin{bmatrix}}
\newcommand{\ebmat}{\end{bmatrix}} 





\title{MAT 250B HW09}
\author{[add your name here]}
\date{Due Friday, 3/8/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle






%%%%%%%%%%%%%%%%%%%%%%
\exercise
Determine whether the following polynomials are irreducible in the rings indicated. For those that are reducible, determine their factorization into irreducibles.
\bea
\item $x^2 + x + 1$ in $\F_2[x]$
\item $x^3 + x + 1$ in $\F_3[x]$
\item $x^4 + 1$ in $\F_5[x]$
\item $x^4 + 10x^2 + 1$ in $\Z[x]$
\item $x^4 - 4x^3 + 6$ in $\Z[x]$
\ee


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Determine the degree of the following elements over $\Q$.
\bea
\item $2 + \sqrt{3}$
\item $1 + \sqrt[3]{2} + \sqrt[3]{4}$
\item $\sqrt{3 + 2 \sqrt{2}}$
\item $\sqrt{1 + \sqrt{-3}} + \sqrt{1 - \sqrt{-3}}$
\ee



%%%%%%%%%%%%%%%%%%%%%%
\exercise
Suppose $F = \Q(\alpha_1, \alpha_2, \ldots, \alpha_n)$ where $\alpha_i^2 \in \Q$ for $i = 1,2,\ldots, n$. Prove that $\sqrt[3]{2} \not\in F$. 

%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that $\Q(\sqrt{2} + \sqrt{3}) = \Q(\sqrt{2},\sqrt{3})$. 
Conclude that $[\Q(\sqrt{2} + \sqrt{3}):\Q] = 4$. Find the minimal polynomial of $\alpha = \sqrt{2} + \sqrt{3}$ over $\Q$.





%%%%%%%%%%%%%%%%%%%%%%
\exercise
Determine the splitting field and its degree over $\Q$ for the following polynomials.
\bea
\item $x^4 - 2$
\item $x^4 + 2$
\item $x^4 + x^2 + 1$
\item $x^6-4$
\ee


%%%%%%%%%%%%%%%%%%%%%%
\exercise
For any prime $p$ and any nonzero $a \in \F_p$, prove that $x^p - x + a$ is irreducible and separable over $\F_p$.
\textit{Hint: Prove that if $\alpha$ is a root then $\alpha+1$ is also a root. }


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that $f(x)^p = f(x^p)$ for any polynomial $f(x) \in \F_p[x]$. 


\end{document}
