Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-33/chapter-33.tex

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+\chapter{Cyclotomic Extensions}
+\subimport{./}{cyclotomic-polynomials.tex}
+\subimport{./}{the-constructible-regular-n-gons.tex}

part-5/chapters/chapter-33/cyclotomic-polynomials.tex

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+\section{Cyclotomic Polynomials}
+
+\begin{remark}
+	Recall from Example 2 in Chapter 16 that the complex zeros of $x^n-1$ are 1, $\omega = \cos(2\pi/n) = i\sin(2\pi/n)$, $\omega^2, \omega^3,\dots,\omega^{n-1}$. Thus, the splitting field of $x^n-1$ over $\Q$ is $\Q(\omega)$. This field is called the \textit{$n$th cyclotomic extension of $\Q$}, and the irreducible factors of $x^n-1$ over $\Q$ are called the \textit{cyclotomic polynomials}.
+
+	Since $\omega=\cos(2\pi/n) + i\sin(2\pi/n)$ generates a cyclic group of order $n$ under multiplication, we know from Corollary 3 of Theorem 4.2 that the generators of $\lr{\omega}$ are the elements of the form $\omega^k$, where $1 \leq k \leq n$ and $\gcd(n,k) = 1$. These generators are called the \textit{primitive $n$th roots of unity}. Recalling that we use $\phi(n)$ to denote the number of positive integers less than or equal to $n$ and relatively prime to $n$, we see that for each positive integer $n$ there are precisely $\phi(n)$ primitive $n$th roots of unity. The polynomials whose zeros are the $\phi(n)$ primitive $n$th roots of unity have a special name.
+\end{remark}
+
+\begin{definition}
+	For any positive integer $n$, let $\omega_1,\omega_2,\dots,\omega_{\phi(n)}$ denote the primitive $n$th roots of unity. the \textit{$n$th cyclotomic polynomial over $\Q$} is the polynomial $\Phi_n(x) = (x-\omega_1)(x-\omega_2)\dots(x-\omega_{\phi(n)})$.
+\end{definition}
+
+\begin{theorem}
+	For every positive integer $n$, $x^n-1 = \Pi_{d\vert n}\Phi_d(x)$, where the product runs over all positive divisors $d$ of $n$.
+\end{theorem}
+
+\begin{theorem}
+	For every positive integer $n$, $\Phi_n(x)$ has integer coefficients.
+\end{theorem}
+
+\begin{theorem}[(Gauss)]
+	The cyclotomic polynomials $\Phi_n(s)$ are irreducible over $\Z$.
+\end{theorem}
+
+\begin{theorem}
+	Let $\omega$ be a primitive $n$th root of unity. Then $\gal(\Q(\omega)/\Q) \approx U(n)$.
+\end{theorem}

part-5/chapters/chapter-33/the-constructible-regular-n-gons.tex

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+\section{The Constructible Regular $\mathbf{n}$-gons}
+
+\begin{lemma}
+	Let $n$ be a positive integer and let $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$. Then $\Q(\cos(2\pi/n)) \subseteq \Q(\omega)$.
+\end{lemma}
+
+\begin{theorem}[(Gauss, 1796)]
+	It is possible to construct the regular $n$-gon with a straightedge and compass if and only if $n$ has the form $2^kp_1p_2\dots p_t$, where $k \geq 0$ and the $p_i$'s are distinct primes of the form $2^m + 1$.
+\end{theorem}