Fix typos in Part II & III

README.md

@@ -3,7 +3,7 @@
 These are my notes for Part II and Part III of Mathematics at the University of Cambridge.
 There are many hyperlinks, marked in blue, to help both learning and revising a topic: I recommend using a PDF viewer with a back function (such as Skim) to make navigation more convenient.
 
-If you would like <img src="https://latex.codecogs.com/gif.latex?\LaTeX" title="\LaTeX" /> help - especially diagrams in Ti*k*Z - feel free to get in touch with me; especially if you're producing open-source notes.
+If you would like $\LaTeX$ help - especially diagrams in Ti*k*Z - feel free to get in touch with me; especially if you're producing open-source notes.
 
 ## Part III notes
 ### Michaelmas

ii/lent/algebraic_geometry.tex

@@ -245,7 +245,7 @@ \subsection{Basic notions}
     \begin{equation*}
         Y_1 = Y_2 \cup Y_2',\quad Y_2 = Y_3 \cup Y_3',\quad \dotsc
     \end{equation*}
-    and so we get an infinite chain of affine varities $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$.
+    and so we get an infinite chain of affine varieties $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$.
     But each $Y_i = Z(I_i)$ for some ideals $I_i$.  Let
     \begin{equation*}
         W = \bigcap Y_i = Z\left(\sum I_i\right) = Z(I)
@@ -939,7 +939,7 @@ \section{Projective space}
     \begin{equation*}
         \begin{tikzcd}
             \left\{\parbox{2cm}{\centering closed subvarieties of $\A^n$}\right\} \rar[leftrightarrow]&
-            \left\{\parbox{5.5cm}{\centering closed subvarities $\overline{Z}$ of $\proj^n$ such that no \hyperlink{def:reducible}{irreducible} component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\
+            \left\{\parbox{5.5cm}{\centering closed subvarieties $\overline{Z}$ of $\proj^n$ such that no \hyperlink{def:reducible}{irreducible} component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\
             Z \rar[mapsto] &\overline{Z} = \text{closure of } \iota(Z) \text{ in } \proj^n
         \end{tikzcd}
     \end{equation*}
@@ -949,7 +949,7 @@ \section{Projective space}
     A projective variety is a closed subvariety of $\proj^n$, some $n$
 \end{defi}
 Recall an \hyperlink{def:affVar}{affine variety} is $k[X] = k[x_1, \dotsc, x_n]/I$, $I = \sqrt{I}$.
-\begin{defi}[Quasivarities]
+\begin{defi}[Quasivarieties]
     A \textbf{quasi-affine variety} is an open subvariety of an \hyperlink{def:affVar}{affine variety}.
     A \textbf{quasi-projective variety} is an open subvariety of a \hyperlink{def:projVar}{projective variety}.
 \end{defi}
@@ -1153,8 +1153,8 @@ \section{Smooth points, dimension, Noether normalisation}
     \end{enumerate}
 \end{eg}
 
-\begin{defi}[Transcendance dimension]\hypertarget{def:trdim}
-    Now we can define $\trdim X$, the \textbf{transcendance dimension} of extension $k \subseteq \hyperlink{def:k(x)}{k(X)}$.
+\begin{defi}[Transcendence dimension]\hypertarget{def:trdim}
+    Now we can define $\trdim X$, the \textbf{transcendence dimension} of extension $k \subseteq \hyperlink{def:k(x)}{k(X)}$.
 \end{defi}
 It is not hard to see $\hyperlink{def:trdim}{\trdim} k(x_1, \dotsc, x_n)/k = \trdim \A^n$ = n. Generally,
 \begin{thm}
@@ -1249,7 +1249,7 @@ \section{Smooth points, dimension, Noether normalisation}
 % new lecture
 To recap:
 Suppose we have affine varieties $X$ and $Y$ with a morphism \begin{equation*}k[X] = k[Y][t]/f(t) \leftarrow k[Y].\end{equation*}
-We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varities $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$.
+We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varieties $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$.
 
 \begin{defi}[Integral extension]\hypertarget{def:intExt}
     $B \subseteq A$ is an \textbf{integral ring extension} if $\forall a \in A$, $\exists$ a monic polynomial $f \in B[t]$ with $f(a) = 0$.
@@ -1290,7 +1290,7 @@ \section{Smooth points, dimension, Noether normalisation}
     and indeed any projection onto a line other than the $x$ or $y$ axis will work.
 \end{eg}
 \begin{thm}
-    If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varities, and $X$, $Y$ irreducible.
+    If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varieties, and $X$, $Y$ irreducible.
     \begin{enumerate}[label=(\alph*)]
         \item $\overline{\varphi(X) = Y} \iff$ algebra homomorphism $k[Y] \to k[X]$ is injective.
         \item Suppose $\overline{\varphi(X) = Y}$. Then
@@ -1335,7 +1335,7 @@ \section{Smooth points, dimension, Noether normalisation}
     \item $x_1, \dotsc, x_n$ generate $A$
     \item for each $i > d$, $x_i$ satisfies a monic irreducible polynomial $F_i$ with coefficients in $k[x_1, \dotsc, x_{i-1}]$.
 \end{enumerate}
-Moreoever, if $k$ is perfect, then $F_i$ can be chosen to be separable.
+Moreover, if $k$ is perfect, then $F_i$ can be chosen to be separable.
 \begin{defi}[Perfect]
     A field $k$ is perfect if $\chara k = p > 0$ and $x \mapsto x^p$ is a surjection.
 \end{defi}
@@ -1511,7 +1511,7 @@ \section{Algebraic Curves}
         =0 \quad \text{when } X = (x_{ij}) \tag{$*$} \label{eq:13star}
     \end{equation*}
     Now recall $X \text{adj}(X) = \det(X) I$ where $\text{adj}(X)$ is the matrix of determinants of minors.
-    Mulitply \eqref{eq:13star} by $\text{adj}(I-X)$, get $d m_i = 0$ for $i = 1, \dotsc, n$ where $d = \det(I-X) = 1+r$ with $r \in J$ by expanding out the determinant, use $J$ ideal i.e.\ $(1+r) M = 0$.
+    Multiply \eqref{eq:13star} by $\text{adj}(I-X)$, get $d m_i = 0$ for $i = 1, \dotsc, n$ where $d = \det(I-X) = 1+r$ with $r \in J$ by expanding out the determinant, use $J$ ideal i.e.\ $(1+r) M = 0$.
 
     (ii) is immediate from (i), by applying (i) to $M/N$.
 \end{proof}
@@ -1535,7 +1535,7 @@ \section{Algebraic Curves}
 \end{remark}
 % ...
 \begin{proof}
-    $X$ is a curve, $\alpha$ defined on an open subset of $X$, so it is defined eexcept possibly at a finite set of points.
+    $X$ is a curve, $\alpha$ defined on an open subset of $X$, so it is defined except possibly at a finite set of points.
     SO it is enough to show $\alpha$ is defined at $p$.
     $Y$ is projective, $Y \subseteq \proj^m$ for some $m$, enough to prove this for $Y = \proj^m$. %(why?)
     \begin{equation*}
@@ -1578,7 +1578,7 @@ \section{Algebraic Curves}
             \begin{equation*}
                 k \hookrightarrow k(Y) \hookrightarrow k(X)
             \end{equation*}
-            but $k \hookrightarrow k(Y)$ has transcedance dimension 1, and $k \hookrightarrow k(X)$ has transcedance dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension.
+            but $k \hookrightarrow k(Y)$ has transcendence dimension 1, and $k \hookrightarrow k(X)$ has transcendence dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension.
     \end{enumerate}
 \end{proof}
 \begin{eg}
@@ -1619,7 +1619,7 @@ \section{Differentials}
 \begin{ex}
     \leavevmode
     \begin{enumerate}[label=(\alph*)]
-        \item let $X$ be an affine algebraic vairety, $x \in X$ and consider the ring homomorphism $\text{ev}_x: k{X} \to k$ given by $f \mapsto f(x)$.
+        \item let $X$ be an affine algebraic variety, $x \in X$ and consider the ring homomorphism $\text{ev}_x: k{X} \to k$ given by $f \mapsto f(x)$.
             Show
             \begin{equation*}
                 \text{Hom}_{k[X]}(\Omega^1_{k[X]/k]}, k) \xrightarrow{\sim} \text{Der}(k[X], k) = T_x X

ii/lent/analysis_of_functions.tex

@@ -263,7 +263,7 @@ \subsection{Lebesgue integration}
     \begin{equation*}\int_E f_k \, d\mu \xrightarrow{k \to \infty} \int_E f \, d\mu.\end{equation*}
 \end{thm}
 \begin{proof}
-    Reduce to $E = X$ by consdering $f_k \chi_E, f \chi_E$.
+    Reduce to $E = X$ by considering $f_k \chi_E, f \chi_E$.
     Then $\left(\int_X f_k \, d\mu\right)_{k \geq 1}$ is a sequence in $[0, \infty]$, non-decreasing.
 
     By monotonicity, $f_k \nearrow f$, so $\int_X f_k \, d\mu \leq \int_X f \, d\mu$.

ii/lent/coding_and_cryptography.tex

@@ -616,7 +616,7 @@ \subsection{Huffman coding}
 \begin{remark}
     Not all \hyperlink{def:optCode}{optimal} codes are \hyperlink{def:huffmanCode}{Huffman}. For instance, take $m=4$, and probabilities $0.3, 0.3, 0.2, 0.2$. An optimal code is given by $00, 01, 10, 11$, but this is not Huffman.
 
-    But, the previous result says that if we have a prefix-free optimal code with word lengths $s_1, \dotsc, s_m$ and associated probabilites $p_1, \dotsc, p_m$, $\exists$ a Huffman code with these word lengths.
+    But, the previous result says that if we have a prefix-free optimal code with word lengths $s_1, \dotsc, s_m$ and associated probabilities $p_1, \dotsc, p_m$, $\exists$ a Huffman code with these word lengths.
 \end{remark}
 
 \begin{defi}[Joint entropy]\hypertarget{def:jointEntropy}
@@ -1972,7 +1972,7 @@ \subsection{Reed-Muller Codes}
 \subsubsection*{Algebraic aside}
 A \textbf{ring} $R$ is a set with two operations, $+$ and $\times$. $(R, +)$ is an additive group and $\times$ is distributive over addition i.e.\ $a(b+c)=ab+ac$. Think of $\Z$ or $\Z_{10}$.
 
-An \textbf{ideal} $I \unlhd R$ is an additive subgroup, closed under external mutliplication, i.e.\ if $a \in I$ and $r \in R$ then $ra \in I$. Think of $2\Z \unlhd Z$.
+An \textbf{ideal} $I \unlhd R$ is an additive subgroup, closed under external multiplication, i.e.\ if $a \in I$ and $r \in R$ then $ra \in I$. Think of $2\Z \unlhd Z$.
 \begin{nlemma}\label{lem:2.27}
     Let $I$ be an ideal in ring $R$ and let $q: R \to R/I$ be the quotient map.
     Then there is a bijection between the set of ideals $J \subseteq R$ containing $I$ and the set of ideals in $R/I$.
@@ -2304,7 +2304,7 @@ \subsubsection*{Decoding BCH codes}
                                &= \sum_{j=1}^\infty e(\alpha^j) X^j \\
         \implies w(X) &= \left(\sum_{j=1}^\infty e(\alpha^j) X^j \right) \sigma(X).
     \end{align*}
-    By defnition of $C$ we have $c(\alpha^j) = 0$ for $1 \leq i \leq \delta-1$ so for $1 \leq j \leq 2t$.
+    By definition of $C$ we have $c(\alpha^j) = 0$ for $1 \leq i \leq \delta-1$ so for $1 \leq j \leq 2t$.
     So $r(\alpha^j) = e(\alpha^j)$ for $1 \leq j \leq 2t$.  Thus
     \begin{equation*}
         \sigma(X) \sum_{j=1}^{2t} r(\alpha^j) X^j \equiv w(X) \pmod{X^{2t+1}}.
@@ -2326,7 +2326,7 @@ \subsubsection*{Decoding BCH codes}
 \end{proof}
 \subsection{Shift registers}
 \begin{defi}[General feedback shift register]\hypertarget{def:fsr}
-    A \textbf{general feedback shift register} is a fucntion $f:\F_2^d \to \F_2^d$ of the form
+    A \textbf{general feedback shift register} is a function $f:\F_2^d \to \F_2^d$ of the form
     \begin{equation*}
         f(x_0, \dotsc, x_{d-1}) = (x_1, \dotsc, x_{d-1}, c(x_0, \dotsc, x_{d-1}))
     \end{equation*}

ii/lent/test.tex

@@ -244,7 +244,7 @@ \subsection{Basic notions}
     \begin{equation*}
         Y_1 = Y_2 \cup Y_2',\quad Y_2 = Y_3 \cup Y_3',\quad \dotsc
     \end{equation*}
-    and so we get an infinite chain of affine varities $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$.
+    and so we get an infinite chain of affine varieties $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$.
     But each $Y_i = Z(I_i)$ for some ideals $I_i$.  Let
     \begin{equation*}
         W = \bigcap Y_i = Z\left(\sum I_i\right) = Z(I)
@@ -938,7 +938,7 @@ \section{Projective space}
     \begin{equation*}
         \begin{tikzcd}
             \left\{\parbox{2cm}{\centering closed subvarieties of $\A^n$}\right\} \rar[leftrightarrow]&
-            \left\{\parbox{5.5cm}{\centering closed subvarities $\overline{Z}$ of $\proj^n$ such that no irreducible component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\
+            \left\{\parbox{5.5cm}{\centering closed subvarieties $\overline{Z}$ of $\proj^n$ such that no irreducible component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\
             Z \rar[mapsto] &\overline{Z} = \text{closure of } \iota(Z) \text{ in } \proj^n
         \end{tikzcd}
     \end{equation*}
@@ -948,7 +948,7 @@ \section{Projective space}
     A projective variety is a closed subvariety of $\proj^n$, some $n$
 \end{defi}
 Recall an \hyperlink{def:affVar}{affine variety} is $k[X] = k[x_1, \dotsc, x_n]/I$, $I = \sqrt{I}$.
-\begin{defi}[Quasivarities]
+\begin{defi}[Quasivarieties]
     A \textbf{quasi-affine variety} is an open subvariety of an \hyperlink{def:affVar}{affine variety}.
     A \textbf{quasi-projective variety} is an open subvariety of a \hyperlink{def:projVar}{projective variety}.
 \end{defi}
@@ -1221,7 +1221,7 @@ \section{Smooth points, dimension, Noether normalisation}
 
 % new lecture
 Suppose we have affine varieties $X$ and $Y$ with morphism $k[X] = k[Y][t]/f(t) \leftarrow k[Y]$.
-We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varities $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$.
+We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varieties $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$.
 
 \begin{defi}
     $B \subseteq A$ is an integral ring extension if $\forall a \in A$, $\exists$ a monic polynomial $f \in B[t]$ with $f(a) = 0$.
@@ -1260,7 +1260,7 @@ \section{Smooth points, dimension, Noether normalisation}
     and indeed any projection onto a line other than the $x$ or $y$ axis will work.
 \end{eg}
 \begin{thm}
-    If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varities, and $X$, $Y$ irreducible.
+    If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varieties, and $X$, $Y$ irreducible.
     \begin{enumerate}[label=(\alph*)]
         \item $\overline{\varphi(X) = Y} \iff$ algebra homomorphism $k[Y] \to k[X]$ is injective.
         \item Suppose $\overline{\varphi(X) = Y}$. Then
@@ -1546,7 +1546,7 @@ \section{Algebraic Curves}
             \begin{equation*}
                 k \hookrightarrow k(Y) \hookrightarrow k(X)
             \end{equation*}
-            but $k \hookrightarrow k(Y)$ has transcedance dimension 1, and $k \hookrightarrow k(X)$ has transcedance dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension.
+            but $k \hookrightarrow k(Y)$ has transcendence dimension 1, and $k \hookrightarrow k(X)$ has transcendence dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension.
     \end{enumerate}
 \end{proof}
 \begin{eg}

ii/mich/algebraic_topology.tex

@@ -1021,7 +1021,7 @@ \subsection{The Galois correspondence}
 \end{defi}
 
 \begin{defi}
-    If $p: \overline{X} \to X$ is a covering space, a covering transofmation or deck transformation is a self-homoemorphism $\phi: \overline{X} \to \overline{X}$ such that $p \circ \phi = p$.
+    If $p: \overline{X} \to X$ is a covering space, a covering transformation or deck transformation is a self-homeomorphism $\phi: \overline{X} \to \overline{X}$ such that $p \circ \phi = p$.
     In particular, $\phi$ is a lift of $p$ along itself, and so if $\overline{X}$ is path connected, $\phi$ is determined by its value at a point.
 
     \begin{equation*}
@@ -1063,7 +1063,7 @@ \subsection{The Galois correspondence}
 
 The next tool is another action of $\pi_1(X, x_0)$.
 Let $p : (\widetilde{X}, \widetilde{x}_0) \to (X, x_0)$ be a universal cover. For each $[\gamma] \in \pi_1(X, x_0)$, we can consider $\widetilde{x}_0 \cdot [\gamma]$ as a different choice of basepoint for $\widetilde{X}$.
-The Lifting Criterion gives a (ubique) covering transofmation $\phi_{[\gamma]} : \widetilde{X} \to \widetilde{X}$.
+The Lifting Criterion gives a (unique) covering transformation $\phi_{[\gamma]} : \widetilde{X} \to \widetilde{X}$.
 
 \begin{lemma}
     This defines a left action of $\pi_1(X, x_0)$ on $\widetilde{X}$ by covering transformation.

ii/mich/galois_theory.tex

@@ -63,7 +63,7 @@ \subsection{Course overview}
 % Galois' papers have been studied by Peter Neumann:
 % The math writings of Evariste Galois, European Math Soc
 % Different books: I. Steward Galois Theory, (something) and Hall
-% contains a historcal introduction and covers almost all the syllabus.
+% contains a historical introduction and covers almost all the syllabus.
 % Artin Galois Theory
 % Van der Waerden Modern Algebra (covers a lot more than Galois theory)
 % Lang Algebra (late editions are preferred, covers a lot of algebra)
@@ -322,7 +322,7 @@ \subsection{Digression on (Non-)Constructibility}
 Schedules mention `other classical problems' and we are now in a position to tackle some of these using \cref{cor:1.11}.
 
 A classical question from Greek geometry concerns the existence or otherwise of constructions using ruler and compasses (where a ruler refers to a single unmarked straight edge).
-If you're an expert you can divide a line betwen 2 points into arbitrarily many equal segments, you can bisect an angle, or you can produce parallel lines.
+If you're an expert you can divide a line between 2 points into arbitrarily many equal segments, you can bisect an angle, or you can produce parallel lines.
 Given a polygon you can produce a square of the same area or double the area. However,
 \begin{enumerate}
     \item You cannot duplicate the cube using ruler and compasses (given a cube you can't produce a cube of double the volume)
@@ -1812,7 +1812,7 @@ \subsection{Galois Theory of Finite Fields}
 \begin{remark}[About \cref{thm:3.13}]
     We'll discover in Number Fields that $\Gal(\overline{f}) \hookrightarrow \Gal(f)$ if $f(t) \in \Z[t]$.
     We factorised $\overline{f}(t) = \overline{g_1}(t) \dotsm \overline{g_s}(t)$, a product of irreducibles.
-    We know from \cref{lem:3.6} that the orbits of $\Gal(\overline{f})$ correspond to the factorsiation.
+    We know from \cref{lem:3.6} that the orbits of $\Gal(\overline{f})$ correspond to the factorisation.
 
     We now know $\Gal(\overline{f})$ is cyclic generated by the Frobenius map, which must have cycle type $(n_1, \dotsc, n_s)$ where $n_j = \deg \overline{g_j}(t)$
 \end{remark}
@@ -2604,7 +2604,7 @@ \subsection{Algebraic closure}
     \begin{equation*}\tilde{f}(t) = f(t) - \prod_{j=1}^{\deg g} (t - X_{f,j}) \in K[X_s: s \in \mathcal{S}][t].\end{equation*}
 
     Let $I \lhd K[X_s : s \in \mathcal{S}]$ generated by all the coefficients of all the $\tilde{f}(t)$.
-    Denote the coefficents of $\tilde{f}(t)$ by $a_{f,l}$ for $0 \leq l \leq \deg f$.
+    Denote the coefficients of $\tilde{f}(t)$ by $a_{f,l}$ for $0 \leq l \leq \deg f$.
 
     Claim: $I \neq K[X_s : s \in \mathcal{S}]$.
     Proof: Suppose $1 \in I$ and aim for a contradiction.

ii/mich/galois_theory_def.tex

@@ -58,7 +58,7 @@ \subsection{Course overview}
 % Galois' papers have been studied by Peter Neumann:
 % The math writings of Evariste Galois, European Math Soc
 % Different books: I. Steward Galois Theory, (something) and Hall
-% contains a historcal introduction and covers almost all the syllabus.
+% contains a historical introduction and covers almost all the syllabus.
 % Artin Galois Theory
 % Van der Waerden Modern Algebra (covers a lot more than Galois theory)
 % Lang Algebra (late editions are preferred, covers a lot of algebra)

ii/mich/galois_theory_thm.tex

@@ -58,7 +58,7 @@ \subsection{Course overview}
 % Galois' papers have been studied by Peter Neumann:
 % The math writings of Evariste Galois, European Math Soc
 % Different books: I. Steward Galois Theory, (something) and Hall
-% contains a historcal introduction and covers almost all the syllabus.
+% contains a historical introduction and covers almost all the syllabus.
 % Artin Galois Theory
 % Van der Waerden Modern Algebra (covers a lot more than Galois theory)
 % Lang Algebra (late editions are preferred, covers a lot of algebra)