Created the Abstract Algebra theorems and definitions cheat sheet
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\chapter{Introduction to Rings}
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\subimport{./}{motivation-and-definition.tex}
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\subimport{./}{properties-of-rings.tex}
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\subimport{./}{subrings.tex}
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\section{Motivation and Definition}
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\begin{definition}[Ring]
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A \textit{ring} $R$ is a set with two binary operations, addition (denoted by $a + b$) and multiplication (denoted by $ab$), such that for all $a,b,c$ in $R$:
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\begin{enumerate}
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\item $a + b = b + a$.
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\item $(a + b) + c = a + (b + c)$.
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\item There is an additive identity 0. That is, there is an element 0 in $R$ such that $a + 0 = a$ for all $a$ in $R$.
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\item There is an element $-a$ in $R$ such that $a + (-a) = 0$.
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\item $a(bc) = (ab)c$.
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\item $a(b+c) = ab + ac$ and $(b + c)a = ba + ca$.
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\end{enumerate}
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\end{definition}
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\begin{remark}
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Note that multiplication need not be commutative. When it is, we say that the ring is \textit{commutative}. Also, a ring need not have an identity under multiplication. A \textit{unity} (or \textit{identity}) in a ring is a nonzero element that is an identity under multiplication. A nonzero element of a com-
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mutative ring with unity need not have a multiplicative inverse. When it does, we say that it is a unit of the ring. Thus, $a$ is a unit if $a^{-1}$ exists.
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\noindent The following terminology and notation are convenient. If $a$ and $b$ belong to a commutative ring $R$ and $a$ is nonzero, we say that $a$ \textit{divides} $b$ (or that $a$ is a \textit{factor} of $b$) and write $a \vert b$, if there exists an element $c$ in $R$ such that $b = ac$. If $a$ does not divide $b$, we write $a \nmid b$.
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\end{remark}
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\section{Properties of Rings}
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\begin{theorem}[Rules of Multiplication]
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Let $a,b$, and $c$ belong to a ring $R$. Then
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\begin{enumerate}
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\item $a0 = 0a = 0$.
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\item $a(-b) = (-a)b = -(ab)$.
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\item $(-a)(-b) = ab$.
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\item $a(b-c) = ab - ac$ and $(b-c)a = ba - ca$.
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\end{enumerate}
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Furthermore, if $R$ has a unity element $1$, then
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\begin{enumerate}
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\setcounter{enumi}{4}
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\item $(-1)a = -a$.
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\item $(-1)(-1) = 1$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[Uniqueness of the Unity and Inverses]
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If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.
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\end{theorem}
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\section{Subrings}
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\begin{definition}[Subring]
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A subset $S$ of a ring $R$ is a \textit{subring of $R$} if $S$ is itself a ring with the operations of $R$.
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\end{definition}
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\begin{theorem}[Subring Test]
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A nonempty subset $S$ of a ring $R$ is a subring if $S$ is closed under subtraction and multiplication -- that is, if $a - b$ and $ab$ are in $S$ whenever $a$ and $b$ are in $S$.
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\end{theorem}
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\chapter{Integral Domains}
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\subimport{./}{definition-and-examples.tex}
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\subimport{./}{fields.tex}
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\subimport{./}{characteristic-of-a-ring.tex}
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\section{Characteristic of a Ring}
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\begin{definition}[Characteristic of a Ring]
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The \textit{characteristic} of a ring $R$ is the least positive integer $n$ such that $nx = 0$ for all $x$ in $R$. If no such integer exists, we say that $R$ has characteristic 0. The characteristic of $R$ is denoted by $\characteristic R$.
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\end{definition}
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\begin{theorem}[Characteristic of a Ring with Unity]
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Let $R$ be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of $R$ is 0. If 1 has order $n$ under addition, then the characteristic of $R$ is $n$.
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\end{theorem}
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\begin{theorem}[Characteristic of an Integral Domain]
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The characteristic of an integral domain is 0 or prime.
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\end{theorem}
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\section{Definition and Examples}
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\begin{definition}[Zero Divisors]
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A \textit{zero-divisor} is a nonzero element $a$ of a commutative ring $R$ such that there is a nonzero element $b \in R$ with $ab = 0$.
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\end{definition}
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\begin{definition}[Integral Domain]
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An \textit{integral domain} is a commutative ring with unity and no zero-divisors.
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\end{definition}
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\begin{theorem}[Cancellation]
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Let $a,b$, and $c$ belong to an integral domain If $a \neq 0$ and $ab = ac$, then $b = c$.
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\end{theorem}
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\section{Fields}
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\begin{definition}[Field]
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A \textit{field} is a commutative ring with unity in which every nonzero element is a unit.
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\end{definition}
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\begin{theorem}[Finite Integral Domains are Fields]
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A finite integral domain is a field.
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\end{theorem}
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\begin{corollary}[$\mathbf{\Z_p}$ Is a Field]
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For every prime $p$, $\Z_p$, the ring of integers modulo $p$ is a field.
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\end{corollary}
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\chapter{Ideals and Factor Rings}
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\subimport{./}{ideals.tex}
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\subimport{./}{factor-rings.tex}
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\subimport{./}{prime-ideals-and-maximal-ideals.tex}
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\section{Factor Rings}
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\begin{theorem}[Existence of Factor Rings]
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Let $R$ be a ring and let $A$ be a subring of $R$. The set of cosets $\{r + A\ \vert\ r \in R\}$ is a ring under the operations $(s + A) + (t + A) = s + t + A$ and $(s+A)(t+A)=st+A$ if and only if $A$ is an ideal of $R$.
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\end{theorem}
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\section{Ideals}
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\begin{definition}[Ideal]
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A subring $A$ of a ring $R$ is called a (two-sided) \textit{ideal} of $R$ if for every $r \in R$ and every $a \in A$ both $ra$ and $ar$ are in $A$.
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\end{definition}
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\begin{theorem}[Ideal Test]
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A nonempty subset $A$ of a ring $R$ is an ideal of $R$ if
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\begin{enumerate}
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\item $a-b \in A$ whenever $a,b \in A$.
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\item $ra$ and $ar$ are in $A$ whenever $a \in A$ and $r \in R$.
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\end{enumerate}
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\end{theorem}
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\section{Prime Ideals and Maximal Ideals}
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\begin{remark}
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A \textit{proper} ideal is an ideal $I$ of some ring $R$ such that it is a proper subset of $R$; that is, $I \subset R$.
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\end{remark}
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\begin{definition}[Prime Ideal, Maximal Ideal]
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A \textit{prime ideal} $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a,b \in R$ and $ab \in A$ imply $a \in A$ or $b \in A$. A \textit{maximal} ideal of a commutative ring $R$ is a \textit{proper} ideal of $R$ such that, whenever $B$ is an ideal of $R$ and $A \subseteq B \subseteq R$, then $B = A$ or $B = R$.
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\end{definition}
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\begin{theorem}[$\mathbf{R/A}$ Is an Integral Domain If and Only If $\mathbf{A}$ Is Prime]
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Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is an integral domain if and only if $A$ is prime.
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\end{theorem}
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\begin{theorem}[$\mathbf{R/A}$ Is a Field If and Only If $\mathbf{A}$ Is Maximal]
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Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is a field if and only if $A$ is maximal.
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\end{theorem}
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\chapter{Ring Homomorphisms}
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\subimport{./}{definition-and-examples.tex}
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\subimport{./}{properties-of-ring-homomorphisms.tex}
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\subimport{./}{the-field-of-quotients.tex}
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\section{Definition and Examples}
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\begin{definition}[Ring Homomorphism, Ring Isomorphism]
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A \textit{ring homomorphism} $\phi$ from a ring $R$ to a ring $S$ is a mapping from $R$ to $S$ that preserves the two ring operations; that is, for all $a,b$ in $R$,
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\[ \phi(a + b) = \phi(a) + \phi(b)\ \ \ \ \text{and}\ \ \ \ \phi(ab) = \phi(a)\phi(b) \]
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A ring homomorphism that is both one-to-one and onto is called a \textit{ring isomorphism}.
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\end{definition}
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\section{Properties of Ring Homomorphisms}
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\begin{theorem}[Properties of Ring Homomorphisms]
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Let $\phi$ be a ring homomorphism from a ring $R$ to a ring $S$. Let $A$ be a subring of $R$ and let $B$ be an ideal of $S$.
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\begin{enumerate}
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\item For any $r \in R$ and any positive integer $n$, $\phi(nr) = n\phi(r)$ and $\phi(r^n) = (\phi(r))^n$.
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\item $\phi(A) = \{\phi(a)\ \vert\ a \in A\}$ is a subring of $S$.
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\item If $A$ is an ideal and $\phi$ is onto $S$, then $\phi(A)$ is an ideal.
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\item $\phi^{-1}(B) = \{r \in R\ \vert\ \phi(r) \in B\}$ is an ideal of $R$.
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\item If $R$ is commutative, then $\phi(R)$ is commutative.
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\item If $R$ has a unity 1, $S \neq \{0\}$, and $\phi$ is onto, then $\phi(1)$ is the unity of $S$.
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\item $\phi$ is an isomorphism if and only if $\phi$ is onto and $\ker \phi = \{r \in R\ \vert\ \phi(r) = 0\} = \{0\}$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[Kernels Are Ideals]
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Let $\phi$ be a ring homomorphism from a ring $R$ to a ring $S$. Then $\ker \phi = \{r \in R\ \vert\ \phi(r) = 0\}$ is an ideal of $R$.
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\end{theorem}
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\begin{theorem}[First Isomorphism Theorem for Rings]
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Let $\phi$ be a ring homomorphism from $R$ to $S$. Then the mapping from $R/\ker \phi$ to $\phi(R)$, given by $r + \ker \phi \to \phi(r)$, is an isomorphism. In symbols, $R/\ker\phi\approx\phi(R)$. This theorem is often referred to as the \textit{Fundamental Theorem of Ring Homomorphisms}.
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\end{theorem}
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\begin{theorem}[Ideals Are Kernels]
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Every ideal of a ring $R$ is the kernel of a ring homomorphism of $R$. In particular, an idea l$A$ is the kernel of the mapping $r \to r + A$ from $R$ to $R/A$. This mapping is known as the \textit{natural homomorphism} from $R$ to $R/A$.
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\end{theorem}
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\begin{theorem}[Homomorphism from $\mathbf{\Z}$ to a Ring with Unity]
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Let $R$ be a ring with unity 1. The mapping $\phi: \Z \to R$ given by $n \to n \cdot 1$ is a ring homomorphism.
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\end{theorem}
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\begin{corollary}[A Ring with Unity Contains $\mathbf{\Z_n}$ or $\mathbf{\Z}$]
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If $R$ is a ring with unity and the characteristic of $R$ is $n > 0$, then $R$ contains a subring isomorphic to $\Z_n$. If the characteristic of $R$ is 0, then $R$ contains a subring isomorphic to $\Z$.
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\end{corollary}
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\begin{corollary}[$\mathbf{\Z_m}$ Is a Homomorphic Image of $\mathbf{\Z}$]
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For any positive integer $m$, the mapping of $\phi: \Z \to \Z_m$ given by $x \to x \mod m$ is a ring homomorphism.
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\end{corollary}
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\begin{corollary}[A Field Contains $\mathbf{\Z_p \text{ or } \Q}$ (Steinitz, 1910)]
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If $\F$ is a field of characteristic $p$, then $\F$ contains a subfield isomorphic to $\Z_p$. If $\F$ is a field of characteristic 0, then $\F$ contains a subfield isomorphic to the rational numbers.
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\end{corollary}
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\section{The Field of Quotients}
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\begin{theorem}[Field of Quotients]
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Let $D$ be an integral domain. Then there exists a field $\F$ (called the field of quotients in $D$) that contains a subring isomorphic to $D$.
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\end{theorem}
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\chapter{Polynomial Rings}
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\subimport{./}{notation-and-terminology.tex}
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\subimport{./}{the-division-algorithm-and-consequences.tex}
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\section{Notation and Terminology}
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\begin{definition}[Ring of Polynomials over $\mathbf{R}$]
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Let $R$ be a commutative ring. The set of formal symbols
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\[ R[x] = \{a_nx^n + a_{n-1}x^{n-1}+\dots+a_1x + a_0\ \vert\ a_i \in R, n \in \Z^+\} \]
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is called the \textit{ring of polynomials over $R$ in the indeterminate $x$}.\\
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\noindent Two elements
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\[ a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
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\noindent and
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\[ b_mx^m + b_{m-1}x^{m-1} + \dots + b_1x + b_0 \]
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\noindent of $R[x]$ are considered equal if and only if $a_i=b_i$ for all nonnegative integers $i$. (Define $a_i=0$ when $i > n$ and $b_i = 0$ when $i > m$.)
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\end{definition}
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\begin{definition}[Addition and Multiplication in $\mathbf{R[x]}$]
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Let $R$ be a commutative ring and let
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\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
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\noindent and
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\[ g(x) = b_mx^m + b_{m-1}x^{m-1} + \dots + b_1x + b_0 \]
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\noindent belong to $R[x]$. Then
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\[ f(x) + g(x) = (a_s + b_s)x^s + (a_{s-1} + b_{s-1})x^{s-1} + \dots + (a_1 + b_1)x + a_0 + b_0 \]
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\noindent where $s$ is the maximum of $m$ and $n$, $a_i = 0$ for $i > n$, and $b_i = 0$ for $i > m$. Also,
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\[ f(x)g(x) = c_{m+n}x^{m+n}+c_{m+n-1}x^{m+n-1} + \dots + c_1x + c_0 \]
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\noindent where
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\[ c_k = a_kb_0 + a_{k-1}b_1 + \dots + a_1b_{k-1} + a_0b_k \]
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\noindent for $k=0,\dots, m+n$.
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\end{definition}
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\begin{theorem}[$\mathbf{D}$ an Integral Domain Implies $\mathbf{D[x]}$ an Integral Domain]
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If $D$ is an integral domain, then $D[x]$ is an integral domain.
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\end{theorem}
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\section{The Division Algorithm and Consequences}
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\begin{theorem}[Division Algorithm for $\mathbf{\F[x]}$]
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Let $\F$ be a field and let $f(x), g(x) \in \F[x]$ with $g(x) \neq 0$. Then there exist unique polynomials $q(x)$ and $r(x)$ in $\F[x]$ such that $f(x) = g(x)q(x) + r(x)$ and either $r(x) = 0$ or $\deg r(x) < \deg g(x)$.
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\end{theorem}
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\begin{corollary}[Remainder Theorem]
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Let $\F$ be a field, $a \in \F$, and $f(x) \in \F[x]$. Then $f(a)$ is the remainder in the division of $f(x)$ by $x -a$.
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\end{corollary}
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\begin{corollary}[Factor Theorem]
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Let $\F$ be a field, $a \in \F$, and $f(x) \in \F[x]$. Then $a$ is a zero of $f(x)$ if and only if $x-a$ is a factor of $f(x)$.
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\end{corollary}
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\begin{corollary}[Polynomials of Degree $\mathbf{n}$ Have at Most $\mathbf{n}$ Zeros]
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A polynomial of degree $n$ over a field has at most $n$ zeros, counting multiplicity.
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\end{corollary}
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\begin{definition}[Principal Ideal Domain (PID)]
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A \textit{principal ideal domain} is an integral domain $R$ in which every ideal has the form $\lr{a}=\{ra\ \vert\ r \in R\}$ for some $a$ in $R$.
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\end{definition}
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\begin{theorem}[$\mathbf{\F[x]}$ Is a PID]
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Let $\F$ be a field. Then $\F[x]$ is a principal ideal domain.
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\end{theorem}
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\begin{theorem}[Criterion for $\mathbf{I = \lr{g(x)}}$]
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Let $\F$ be a field, $I$ a nonzero ideal in $\F[x]$, and $g(x)$ an element of $\F[x]$. Then, $I=\lr{g(x)}$ if and only if $g(x)$ is a nonzero polynomial of minimum degree in $I$.
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\end{theorem}
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\chapter{Factorization of Polynomials}
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\subimport{./}{reducibility-tests.tex}
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\subimport{./}{irreducibility-tests.tex}
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\subimport{./}{unique-factorization-in-zx.tex}
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\section{Irreducibility Tests}
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\begin{theorem}[Mod $\mathbf{p}$ Irreducibility Test]
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Let $p$ be a prime and suppose that $f(x) \in \Z[x]$ with $\deg f(x) \geq 1$. Let $\overline{f}(x)$ be the polynomial in $\Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ modulo $p$. If $\overline{f}(x)$ is irreducible over $\Z_p$ and $\deg \overline{f}(x) = \deg f(x)$, then $f(x)$ is irreducible over $\Q$.
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\end{theorem}
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\begin{theorem}[Eisenstein's Criterion (1850)]
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Let
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\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 \in \Z[x] \]
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\noindent If there is a prime $p$ such that $p \nmid a_n, p\ \vert\ a_{n-1}, \dots, p\ \vert\ a_0$ and $p^2 \nmid a_0$, then $f(x)$ is irreducible over $\Q$.
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\end{theorem}
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\begin{corollary}[Irreducibility of $\mathbf{p}$th Cyclotomic Polynomial]
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For any prime $p$, the $p$th cyclotomic polynomial
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\[ \Phi_p(x) = \frac{x^p - 1}{x-1} = x^{p-1} + x^{p-2} + \dots + x + 1 \]
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\noindent is irreducible over $\Q$.
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\end{corollary}
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\begin{theorem}[$\mathbf{\lr{p(x)}}$ Is Maximal If and Only If $\mathbf{p(x)}$ Is Irreducible]
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Let $\F$ be a field and let $p(x) \in \F[x]$. Then $\lr{p(x)}$ is a maximal ideal in $\F[x]$ if and only if $p(x)$ is irreducible over $\F$.
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\end{theorem}
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\begin{corollary}[$\mathbf{\F[x]/\lr{p(x)}}$ Is a Field]
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Let $\F$ be a field and $p(x)$ be an irreducible polynomial over $\F$. Then $\F[x]/\lr{p(x)}$ is a field.
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\end{corollary}
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\begin{corollary}[$\mathbf{p(x)\ \vert\ a(x)b(x)}$ Implies $\mathbf{p(x)\ \vert\ a(x)}$ or $\mathbf{p(x)\ \vert\ b(x)}$]
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Let $\F$ be a field and let $p(x), a(x), b(x) \in \F[x]$. If $p(x)$ is irreducible over $\F$ and $p(x)\ \vert\ a(x)b(x)$, then $p(x)\ \vert\ a(x)$ or $p(x)\ \vert\ b(x)$.
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\end{corollary}
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\section{Reducibility Tests}
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\begin{definition}[Irreducible Polynomial, Reducible Polynomial]
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||||
Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be \textit{irreducible over $D$}, whenever $f(x)$ is expressed as a product $f(x) = g(x)h(x)$, with $g(x)$ and $h(x)$ from $D[x]$, then $g(x)$ or $h(x)$ is a unit in $D[x]$. A nonzero, nonunit element of $D[x]$ that is not irreducible over $D$ is called \textit{reducible over $D$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Reducibility Test for Degrees 2 and 3]
|
||||
Let $\F$ be a field. If $f(x) \in \F[x]$ and $\deg f(x)$ is 2 or 3, then $f(x)$ is reducible over $\F$ if and only if $f(x)$ has a zero in $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Content of a Polynomial, Primitive Polynomial]
|
||||
The \textit{content} of a nonzero polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_0$, where the $a$'a are integers, is the greatest common divisor of the integers $a_n,a_{n-1}, \dots, a_0$. A \textit{primitive polynomial} is an element of $\Z[x]$ with content 1.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}[Gauss's Lemma]
|
||||
The product of two primitive polynomials is primitive.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Reducibility over $\mathbf{\Q}$ Implies Reducibility over $\mathbf{\Z}$]
|
||||
Let $f(x) \in \Z[x]$. If $f(x)$ is reducible over $\Q$, then it is reducible over $\Z$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,7 @@
|
||||
\section{Unique Factorization In $\mathbf{\Z[x]}$}
|
||||
|
||||
\begin{theorem}[Unique Factorization in $\mathbf{\Z[x]}$]
|
||||
Every polynomial in $\Z[x]$ that is not the zero polynomial or a unit in $\Z[x]$ can be written in the form $b_1b_2\dots b_sp_1(x)p_2(x)\dots p_m(x)$, where the $b_i$'s are irreducible polynomials of degree 0 and the $p_i(x)$'s are irreducible polynomials of positive degree. Furthermore, if
|
||||
\[ b_1b_2\dots b_sp_1(x)p_2(x) \dots p_m(x) = c_1c_2 \dots c_tq_1(x) q_2(x) \dots q_n(x) \]
|
||||
\noindent where the $b_i$'s and the $c_i$'s are irreducible polynomials of degree 0 and the $p_i(x)$'s and $q_i(x)$'s are irreducible polynomials of positive degree, then $s=t, m=n$, and, after renumbering the $c$'s and $q(x)$'s, we have $b_i = \pm c_i$, for $i=1, \dots, s$, and $p_i(x)= \pm q_i(x)$, for $i = 1, \dots, m$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Divisibility in Integral Domains}
|
||||
\subimport{./}{irreducibles-primes.tex}
|
||||
\subimport{./}{unique-factorization-domains.tex}
|
||||
\subimport{./}{euclidean-domains.tex}
|
||||
@@ -0,0 +1,21 @@
|
||||
\section{Euclidean Domains}
|
||||
|
||||
\begin{definition}[Euclidean Domain (ED)]
|
||||
An integral domain $D$ is called a \textit{Euclidean domain} if there is a function $d$ (called the \textit{measure}) from nonzero elements of $D$ to the nonnegative integers such that
|
||||
\begin{enumerate}
|
||||
\item $d(a) \leq d(ab)$ for all nonzero $a,b \in D$; and
|
||||
\item if $a,b \in D,\ b \neq 0$, then there exist elements $q$ and $r$ in $D$ such that $a = bq + r$, where $r = 0$ or $d(r) < d(b)$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[ED Implies PID]
|
||||
Every Euclidean domain is a principal ideal domain.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[ED Implies UFD]
|
||||
Every Euclidean domain is a unique factorization domain.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[$\mathbf{D}$ a UFD Implies $\mathbf{D[x]}$ a UFD]
|
||||
If $D$ is a unique factorization domain, then $D[x]$ is a unique factorization domain.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Irreducibles, Primes}
|
||||
|
||||
\begin{definition}[Associates, Irreducibles, Primes]
|
||||
Elements $a$ and $b$ of an integral domain $D$ are called \textit{associates} if $a = ub$, where $u$ is a unit of $D$. A nonzero element $a$ of an integral domain $D$ is called an \textit{irreducible} if $a$ is not a unit and, whenever $b$, $c \in D$ with $a = bc$, then $b$ or $c$ is a unit. A nonzero element $a$ of an integral domain $D$ is called a \textit{prime} if $a$ is not a unit and $a\ \vert\ bc$ implies $a\ \vert\ b$ or $a\ \vert\ c$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Prime Implies Irreducible]
|
||||
In an integral domain, every prime in an irreducible.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[PID Implies Irreducible Equals Prime]
|
||||
In a principal ideal domain, an element is an irreducible if and only if it is a prime.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,21 @@
|
||||
\section{Unique Factorization Domains}
|
||||
|
||||
\begin{definition}
|
||||
An integral domain $D$ is a \textit{unique factorization domain} if
|
||||
\begin{enumerate}
|
||||
\item every nonzero element of $D$ that is not a unit can be written as a product of irreducibles of $D$; and
|
||||
\item the factorization into irreducibles is unique up to associates and the order in which the factors appear.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}[Ascending Chain Condition for a PID]
|
||||
In a principal ideal domain, any stricly increasing chain of ideals $I_1 \subset I_2 \subset \dots$ must be finite in length.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[PID Implies UFD]
|
||||
Every principal ideal domain is a unique factorization domain.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\F[x]}$ Is a UFD]
|
||||
Let $\F$ be a field. Then $\F[x]$ is a unique factorization domain.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,8 @@
|
||||
\part{Rings}
|
||||
\subimport{chapters/chapter-12/}{chapter-12.tex}
|
||||
\subimport{chapters/chapter-13/}{chapter-13.tex}
|
||||
\subimport{chapters/chapter-14/}{chapter-14.tex}
|
||||
\subimport{chapters/chapter-15/}{chapter-15.tex}
|
||||
\subimport{chapters/chapter-16/}{chapter-16.tex}
|
||||
\subimport{chapters/chapter-17/}{chapter-17.tex}
|
||||
\subimport{chapters/chapter-18/}{chapter-18.tex}
|
||||
Reference in New Issue
Block a user