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docs/math/abstract-algebra/16-rings.md

@@ -155,13 +155,13 @@ $$
 \end{align}
 $$
 
-**Theorem**. Chineese Remainer Theorem. Let $n_1, \ldots, n_k \in \mathbb{N}$ be given such that $\gcd(n_i, n_j) = 1$. Then, for any integers $a_1, \ldots, a_k$, the system
+**Theorem**. Chinese Remainder Theorem. Let $n_1, \ldots, n_k \in \mathbb{N}$ be given such that $\gcd(n_i, n_j) = 1$. Then, for any integers $a_1, \ldots, a_k$, the system
 
 $$
 \begin{align}
     x &\equiv a_1 \pmod{n_1} \\
     x &\equiv a_2 \pmod{n_2} \\
-    \vdots
+    \ldots
     x &\equiv a_k \pmod{n_k}
 \end{align}
 $$

docs/math/abstract-algebra/18-integral-domains.md

@@ -53,7 +53,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
 
 **Lemma**. 18.14: Let $D$ be a PID. Let $I_1 \subseteq I_2 \subseteq \ldots$. Then, there exists some integer $N$ such that $I_n = I_N$ for all $n > N$. That is, any chain of ideals converges.
 
-**Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherien ring*.
+**Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherian ring*.
 
 **Theorem**. 18.15: Every PID is a UFD. Note that the converse is not true.
 

docs/math/abstract-algebra/DF-10-modules.md

@@ -5,7 +5,7 @@
 **Definition**. Let $R$ be a ring. A *left $R$-module* or a *left module over $R$* is a nonempty set $M$ together with
 
 1. A binary operation $+$ on $M$ under which $M$ is an abelian group
-2. An action $\cross$ of $R$ on $M$, that is, a map or function $R \cross M \rightarrow M$, denoted $rm$, that for all $r, s \in R, m, n \in M$ satisfies
+2. An action $\times$ of $R$ on $M$, that is, a map or function $R \times M \rightarrow M$, denoted $rm$, that for all $r, s \in R, m, n \in M$ satisfies
     - $(r + s)m = rm + sm$
     - $(rs)m = r(sm)$
     - $r(m + n) = rm + rn$
@@ -21,12 +21,12 @@
 
 ---
 
-**Example**. Let $F$ be a field and $F[x]$ a polynomial ring. Then, let $V$ be a vector space of $F$, and $T$ be a linear transformation from $V$ to itself. That is, $V: T \rightarrow T$. We know that $V$ is an $F$-module. We will want to show that $V$ can be written as an $F[x]$-module for some choice of $T$. That is, we want an action $F[x] \cross V \rightarrow V$.
+**Example**. Let $F$ be a field and $F[x]$ a polynomial ring. Then, let $V$ be a vector space of $F$, and $T$ be a linear transformation from $V$ to itself. That is, $V: T \rightarrow T$. We know that $V$ is an $F$-module. We will want to show that $V$ can be written as an $F[x]$-module for some choice of $T$. That is, we want an action $F[x] \times V \rightarrow V$.
 
-Now, for a given linear transformation $T$, consider some polynomial $p(x) = a_n x^n + \ldots + a_0$ and some $v \in V$. We define $p(x) \cross v$ by$
+Now, for a given linear transformation $T$, consider some polynomial $p(x) = a_n x^n + \ldots + a_0$ and some $v \in V$. We define $p(x) \times v$ by$
 
 $$
-p(x) \cross v = a_n T^n(v) + a_{n-1} T^{n-1}(v) + \ldots + a_0 v
+p(x) \times v = a_n T^n(v) + a_{n-1} T^{n-1}(v) + \ldots + a_0 v
 $$
 
 with $T^n$ being defined as applying $T$ a total of $n$ times.
@@ -50,20 +50,20 @@ with $T^n$ being defined as applying $T$ a total of $n$ times.
 
 **Theorem**. An $R$-module homomorphism is an *isomorphism* if it is 1-1 and onto, and said modules are *isomorphic*.
 
-**Definition**. Let $M, N$ be $R$-modules. The set $\Hom_R(M, N)$ is the set of all homomorphisms from $M$ to $N$.
+**Definition**. Let $M, N$ be $R$-modules. The set $\text{Hom}_R(M, N)$ is the set of all homomorphisms from $M$ to $N$.
 
 **Proposition**. Let $M$, $N$, and $L$ be $R$-modules. Then,
 
 1. A function $\varphi: M \rightarrow N$ is an $R$-module homomorphism if and only if $\varphi(rx + y) = r\varphi(x) + \varphi(y)$ for all $x, y \in M$ and $r \in R$.
-2. Let $\varphi, \psi \in \Hom_R(M, N)$. Then, define $\varphi + \psi$ as
+2. Let $\varphi, \psi \in \text{Hom}_R(M, N)$. Then, define $\varphi + \psi$ as
 
 $$
 (\varphi + \psi)(m) = \varphi(m) + \psi(m)
 $$
 
-Then, $\varphi + \psi \in \Hom_R(M, N)$. Additionally, if $R$ is commutative, with $(r\varphi)(m) = r(\varphi(m))$, then $r\varphi \in \Hom_R(M,N)$
-3. If $\varphi \in \Hom_R(L, M)$ and $\psi \in \Hom_R(M, N)$, then $\psi \circ \varphi \in \Hom_R(L, N)$
-4. $\Hom_R(M, M)$ is a ring with identity. With $R$ being commutative, $\Hom_R(M, M)$ is an $R$-algebra.
+Then, $\varphi + \psi \in \text{Hom}_R(M, N)$. Additionally, if $R$ is commutative, with $(r\varphi)(m) = r(\varphi(m))$, then $r\varphi \in \text{Hom}_R(M,N)$
+3. If $\varphi \in \text{Hom}_R(L, M)$ and $\psi \in \text{Hom}_R(M, N)$, then $\psi \circ \varphi \in \text{Hom}_R(L, N)$
+4. $\text{Hom}_R(M, M)$ is a ring with identity. With $R$ being commutative, $\text{Hom}_R(M, M)$ is an $R$-algebra.
 
 **Proposition**. Let $R$ be a ring, $M$ an $R$-module, and $N \subseteq M$ an $R$-submodule. then, $M/N$ can be made into an $R$-module by defining addition. With $r \in R$ and $x + N \in M/N$,
 

docs/math/abstract-algebra/DF-12-modules-pids.md

@@ -25,22 +25,22 @@ there exists some $k \in \mathbb{N}$ such that given any $n \in \mathbb{N}$ with
 **Definition**. Given $R$ an integral domain and $M$ an $R$-module,
 
 $$
-\Tor(M) = \{ x \in M | rx = 0 \text{ for any } r \neq 0 \}
+\text{Tor}(M) = \{ x \in M | rx = 0 \text{ for any } r \neq 0 \}
 $$
 
-This is the *torsion submodule* of $M$. If $\Tor(M)$ is empty, then $M$ is *torsion-free*.
+This is the *torsion submodule* of $M$. If $\text{Tor}(M)$ is empty, then $M$ is *torsion-free*.
 
 **Definition** Let $R$ be an integral domain and $M$ be an $R$-module. Then, given a submodule $N$,
 
 $$
-\Ann_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \}
+\text{Ann}_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \}
 $$
 
-This ideal of $R$ is the *annihilator of $N$*. That is, $\Ann(N)$ is the set of elements of $R$ such that $(r)N = \{ 0 \}$.
+This ideal of $R$ is the *annihilator of $N$*. That is, $\text{Ann}(N)$ is the set of elements of $R$ such that $(r)N = \{ 0 \}$.
 
-Note that if $N$ is not a torsion submodule of $M$, then $\Ann(N) = (0)R$. Additionally, given $N, L$ are submodules of $M$ with $N \subseteq L$, then $\Ann(N) \subseteq \Ann(L)$.
+Note that if $N$ is not a torsion submodule of $M$, then $\text{Ann}(N) = (0)R$. Additionally, given $N, L$ are submodules of $M$ with $N \subseteq L$, then $\text{Ann}(N) \subseteq \text{Ann}(L)$.
 
-Additionally, if $R$ is a PID, as $\Ann_R(N)$ is an ideal, $\Ann(N) = (n)R$ and $\Ann(L) = (l)R$ for some $n, l \in R$ such that $n | l$.
+Additionally, if $R$ is a PID, as $\text{Ann}_R(N)$ is an ideal, $\text{Ann}(N) = (n)R$ and $\text{Ann}(L) = (l)R$ for some $n, l \in R$ such that $n | l$.
 
 **Definition**. Given any integral domain $R$, the *rank* of an $R$-module $M$ is the maximum number of $R$-linearly independent elements of M.
 
@@ -64,7 +64,7 @@ $$
 - Note that
 
 $$
-\Tor{M} \cong \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}
+\text{Tor}{M} \cong \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}
 $$
 
 As a consequence, $M$ is a torsion module if and only if $r = 0$.
@@ -79,4 +79,4 @@ $$
 
 with $p_t$ non-unique primes and $\alpha_t$ non-unique, but with $(p_t^{\alpha_t})$ unique. These are called the *elementary divisors* of $M$.
 
-TODO: Incomplete for Now
+TODO: Incomplete for Now

docs/math/abstract-algebra/DF-13-fields.md

@@ -1,8 +1,8 @@
-# Dummit & Foote Chapter 12 - Field Theory
+# Dummit & Foote Chapter 13 - Field Theory
 
 ## Section 13.1 Basic Theory of Field Extensions
 
-**Definition**. The *charactaristic* of a field $F$ is the smallest positive integer $p$ such that $1_F * p = 0$. It follows that $p$ is $0$ or prime, and $p \alpha = 0$ for any $\alpha \in F$.
+**Definition**. The *characteristic* of a field $F$ is the smallest positive integer $p$ such that $1_F * p = 0$. It follows that $p$ is $0$ or prime, and $p \alpha = 0$ for any $\alpha \in F$.
 
 **Definition**. If $K, F$ are fields such that $F \subseteq K$, then $K$ is an *extension field* or *extension* of $F$, denoted $K / F$.