:/

docs/math/abstract-algebra/17-polynomial-rings.md

@@ -36,9 +36,9 @@ $$
 
 where either $\deg r(x) < \deg g(x)$ or $r(x)$ is the zero polynomial.
 
-**Collary**. Let $F$ be a field. Then, an element $\alpha \in F$ is a zero of $p(x) \ in F[x]$ if and only if $(x-\alpha)$ is a factor of $p(x)$.
+**Corollary**. Let $F$ be a field. Then, an element $\alpha \in F$ is a zero of $p(x) \ in F[x]$ if and only if $(x-\alpha)$ is a factor of $p(x)$.
 
-**Collary**. Let $F$ be a field. Then, a nonzero polynomial $p(x) \in F[x]$ with degree $n$ can have at most $n$ distinct zeros in $F$.
+**Corollary**. Let $F$ be a field. Then, a nonzero polynomial $p(x) \in F[x]$ with degree $n$ can have at most $n$ distinct zeros in $F$.
 
 **Definition**. A monic polynomial $d(x)$ is the *greatest common divisor* of polynomials $p(x), q(x) \in F[x]$ if $d(x)$ evenly divides both $p(x)$ and $q(x)$. We write $\gcd(p(x), q(x)) = d(x)$. This polynomial is unique.
 
@@ -52,7 +52,7 @@ where either $\deg r(x) < \deg g(x)$ or $r(x)$ is the zero polynomial.
 
 **Lemma**. Gauss's Lemma. Let $p(x) \in \mathbb{Z}[x]$ be monic such that $p(x)$ factors into two polynomials $\alpha(x), \beta{x} \in \mathbb{Q}[x]$, with the degrees of both strictly less than the degree of $p(x)$. Then, there exists two polynomials $a(x), b(x) \in \mathbb{Z}[x]$ such that $p(x) = a(x)b(x)$, and $\deg \alpha(x) = \deg a(x)$ and $\deg \beta(x) = \deg b(x)$.
 
-**Collary**. Let $p(x) \in \mathbb{Z}[x]$ be monic with constant term $a_0$. Then, if $p(x)$ has a zero in $\mathbb{Q}$, then it also has a zero $\alpha$ in $\mathbb[Z]$. Furthermore, $\alpha$ divides $a_0$.
+**Corollary**. Let $p(x) \in \mathbb{Z}[x]$ be monic with constant term $a_0$. Then, if $p(x)$ has a zero in $\mathbb{Q}$, then it also has a zero $\alpha$ in $\mathbb[Z]$. Furthermore, $\alpha$ divides $a_0$.
 
 **Theorem**. Eisenstein's Criterion. Let $p$ be prime, and suppose that
 

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

@@ -18,9 +18,9 @@ $$
 
 Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, there exists a map $\psi: F_D \rightarrow D$ giving an isomorphism such that $\psi(a) = a$ for all $a \in D$.
 
-**Collary**. 18.6: Let $F$ be a field of charactaristic $0$. Then, $F$ contains a subfield isomorphic to $\mathbb{Q}$.
+**Corollary**. 18.6: Let $F$ be a field of charactaristic $0$. Then, $F$ contains a subfield isomorphic to $\mathbb{Q}$.
 
-**Collary**. 18.6: Let $F$ be a field of charactaristic $p$. Then, $F$ contains a subfield isomorphic to $\mathbb{Z}_p$.
+**Corollary**. 18.6: Let $F$ be a field of charactaristic $p$. Then, $F$ contains a subfield isomorphic to $\mathbb{Z}_p$.
 
 ## Section 18.2 - Factorization in Integral Domains
 
@@ -49,7 +49,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
 
 **Theorem**. 18.12: Let $D$ be a PID, and let $\langle p \rangle$ be a nonzero ideal in $D$. Thus, $\langle p \rangle$ is a maximal ideal if and only if $p$ is irreducible.
 
-**Collary**. 18.13: Let $D$ be a PID. For any $p \in D$, if $p$ is irreducible, then $p$ is prime.
+**Corollary**. 18.13: Let $D$ be a PID. For any $p \in D$, if $p$ is irreducible, then $p$ is prime.
 
 **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.
 
@@ -57,7 +57,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
 
 **Theorem**. 18.15: Every PID is a UFD. Note that the converse is not true.
 
-**Collary** 18.16: Let $F$ be a field. Then, $F[x]$ is a UFD.
+**Corollary** 18.16: Let $F$ be a field. Then, $F[x]$ is a UFD.
 
 ---
 
@@ -74,7 +74,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
 
 **Theorem**. 18.21: Every Euclidian domain is a PID.
 
-**Collary**. Every Euclidian domain is a UFD.
+**Corollary**. Every Euclidian domain is a UFD.
 
 ---
 
@@ -88,13 +88,13 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
 
 As a direct consequence, we see the following.
 
-**Collary**. Let $D$ be a UFD, and $F = F_D$. Then, a primitive polynomial $p(x) \in D[x]$ is irreducible in $D[x]$ if and only if it is irreducible in $F[x]$.
+**Corollary**. Let $D$ be a UFD, and $F = F_D$. Then, a primitive polynomial $p(x) \in D[x]$ is irreducible in $D[x]$ if and only if it is irreducible in $F[x]$.
 
-**Collary**. Let $D$ be a UDF, and $F = F_D$. Then, if a monic polynomial $p(x) \ in D[x]$ can be written as $p(x) = f(x)g(x)$ with $f(x), g(x) \in F_D[x]$, then $p(x)$ can be written as $p(x) = f_1(x)g_1(x)$, where $f_1(x), g_1(x) \in D[x]$.
+**Corollary**. Let $D$ be a UDF, and $F = F_D$. Then, if a monic polynomial $p(x) \ in D[x]$ can be written as $p(x) = f(x)g(x)$ with $f(x), g(x) \in F_D[x]$, then $p(x)$ can be written as $p(x) = f_1(x)g_1(x)$, where $f_1(x), g_1(x) \in D[x]$.
 
 **Theorem**. If $D$ is as UFD, then $D[x]$ is a UFD.
 
-**Collary**. This theorem has several collaries:
+**Corollary**. This theorem has several collaries:
 
 1. Given a field $F$, since $F$ is a PID, it is also a UFD. Thus, $F[x]$ is a UFD.
 2. The ring of polynomials over integers, $\mathbb{Z}[x]$ is a UFD.

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

@@ -120,9 +120,9 @@ This direct product is in itself an $R$-module.
 
 **Theorem**. For any set $A$, there is a free $R$-module $F(A)$ on $A$ such that $F(A)$ satisfies the universal property: if $M$ is any $R$-module, and $\varphi: A \rightarrow M$ is a map of sets, there eixts a unique $R$-module homomorphism: $\Phi: F(A) \rightarrow M$ such that $\Phi(a) = \varphi(a)$ for all $a \in A$.
 
-**Collary**. If $F_1$ and $F_2$ are free modules on $A$, then there is a unique isomorphism between $F_1$ and $F_2$, which is the identity map on A.
+**Corollary**. If $F_1$ and $F_2$ are free modules on $A$, then there is a unique isomorphism between $F_1$ and $F_2$, which is the identity map on A.
 
-**Collary**. If $F$ is a free $R$-module with basis $A$, then $F \cong F(A)$.
+**Corollary**. If $F$ is a free $R$-module with basis $A$, then $F \cong F(A)$.
 
 **Definition** For a free module $F$ with basis $A$, if $R$ is commutative, tthen the *rank* of $F$ is the cardinality of $A$.
 

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

@@ -18,7 +18,7 @@ there exists some $k \in \mathbb{N}$ such thaht given any $n \in \mathbb{N}$ wit
 2. Every nonempty set of submodules of $M$ contains a maximal element under inclusion
 3. Every submodule of $M$ is finitely-generated
 
-**Collary**. If $R$ is a principal ideal domain (PID), then all nonempty set of ideals of $R$ has a maximal element. Additionally, $R$ is as Noetherian ring.
+**Corollary**. If $R$ is a principal ideal domain (PID), then all nonempty set of ideals of $R$ has a maximal element. Additionally, $R$ is as Noetherian ring.
 
 **Proposition**. Let $R$ be an integral doman, and $M$ be a free $R$-module of rank $n < \infty$. Then, given $S$ is subset $M$ with $|S| > n$, the elements of $S$ are $R$-linearly dependent.
 
@@ -44,7 +44,7 @@ Additionally, if $R$ is a PID, as $\Ann_R(N)$ is an ideal, $\Ann(N) = (n)R$ and
 
 **Definition**. Given any integral domain $R$, the *rank* of an $R$-module $M$ is the maximum number of $R$-linearly independent elements of M.
 
-**Collary**. The rank of a free module is the number of generating elements.
+**Corollary**. The rank of a free module is the number of generating elements.
 
 **Theorem**. Let $R$ be a principal ideal domain, and $M$ be a free $R$-module of finite rank $m$, and $N$ be a submodule of $M$. Then,
 

docs/math/real-analysis/2-reals.md

@@ -56,7 +56,7 @@
 2. $|ab| = |a||b|$
 3. $|a + b| \leq |a| + |b|$
 
-**Collary**. Givem $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$.
+**Corollary**. Givem $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$.
 
 **Remark**. Every field has at least one absolute value function.
 
@@ -104,4 +104,4 @@ $$
 
 **Theorem**. If $a < b$, then the interval $[a, b]$ is an uncountable set.
 
-**Collary**. $\mathbb{R}$ is uncountable.
+**Corollary**. $\mathbb{R}$ is uncountable.

docs/math/real-analysis/3-sequences-series.md

@@ -96,7 +96,7 @@ is a *subsequence* of $X$,
 
 **Theorem**. Every sequence of real numbers $(x_n)$ contains a monotonic subsequence $(x_{n_k})$.
 
-**Collary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
+**Corollary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
 
 ## Section 3.5 - The Cauchy Criterion
 
@@ -136,9 +136,9 @@ $$
 \abs{s_m - s_n} = \abs{\sum_{i = m + 1}^n x_i} < \epsilon
 $$
 
-**Collary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
+**Corollary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
 
-**Collary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
+**Corollary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
 
 ---
 

docs/math/real-analysis/4-limits.md

@@ -6,11 +6,11 @@
 
 **Theorem**. A real number $c$ is a cluster point for a set $A$ if and only if there exists a sequence $(a_n)$ in $A\\ \{c\}$ such that $a_n \rightarrow c$
 
-**Collary**. A real number $c$ is a cluster point of a set $A$ if and only if every $\delta$-neighborhood conttains infinitely many points of $A$.
+**Corollary**. A real number $c$ is a cluster point of a set $A$ if and only if every $\delta$-neighborhood conttains infinitely many points of $A$.
 
 **Definition**. The set of every cluster point of $A$ is called the *derived set* of $A$, and denoted $A'$.
 
-**Collary**. A set $A$ is closed if and only if $A' \subseteq A$.
+**Corollary**. A set $A$ is closed if and only if $A' \subseteq A$.
 
 **Remark**. If $A'$ is the derived set of $A$, then $A'' \subseteq A'$.
 
@@ -58,7 +58,7 @@ $$
 \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{L}{M}
 $$
 
-**Collary**. If $p, q \in \mathbb{R}[x]$, and $q(c) \neq 0$ for some $c \in \mathbb{R}$, then
+**Corollary**. If $p, q \in \mathbb{R}[x]$, and $q(c) \neq 0$ for some $c \in \mathbb{R}$, then
 
 $$
 \lim_{x \rightarrow c} p(x) = p(c)

docs/math/real-analysis/5-continuity.md

@@ -43,21 +43,21 @@ $$
 \lim_{x \rightarrow c} g(f(x)) = g(L) = g(\lim_{x \rightarrow c} f(x))
 $$
 
-**Collary**. let $A, B \subseteq  \mathbb{R}$, with $f: A \rightarrow B$ and $g: B \rightarrow \mathbb{R}$. If $f$ is continuous at $a \in A$ and $g$ is continuous at $f(a) \in B$, then $g(f(x))$ is continuous at $a$.
+**Corollary**. let $A, B \subseteq  \mathbb{R}$, with $f: A \rightarrow B$ and $g: B \rightarrow \mathbb{R}$. If $f$ is continuous at $a \in A$ and $g$ is continuous at $f(a) \in B$, then $g(f(x))$ is continuous at $a$.
 
 ## Section 5.3 - continuous functions on Intervals
 
 **Theorem**. Let $S, T$ be metric spaces with $A \subseteq S$ and $f: A \rightarrow T$. If $A$ is a compact subset of $S$, then $f(A)$ is a compact subset of $T$.
 
-**Collary**. Let $f: A \rightarrow \mathbb{R}$ be a continuous function, with $A$ being a compact subset of metric space $S$. Then, $f(A)$ is closed and bounded. Moreover, there exists a $p, q \in A$ such that $f(p)$ and $f(q)$ are the supremum and infimum of $f(A)$.
+**Corollary**. Let $f: A \rightarrow \mathbb{R}$ be a continuous function, with $A$ being a compact subset of metric space $S$. Then, $f(A)$ is closed and bounded. Moreover, there exists a $p, q \in A$ such that $f(p)$ and $f(q)$ are the supremum and infimum of $f(A)$.
 
-**Collary**. Maximum-Minimum Theorem. If $I = [a, b]$ is a closed and bounded interval and $f: I \rightarrow \mathbb{R}$ is continuous on $I$, then $f$ has an absolute minumum and maximum on $I$.
+**Corollary**. Maximum-Minimum Theorem. If $I = [a, b]$ is a closed and bounded interval and $f: I \rightarrow \mathbb{R}$ is continuous on $I$, then $f$ has an absolute minumum and maximum on $I$.
 
 ---
 
 **Theorem**. Let $S, T$ be metric spaces and $A \subseteq S$. Then, if $f: A \rightarrow T$ is continuous on $A$, and $A$ is a connected subset of $S$, then $f(A)$ is a connected subset of $T$.
 
-**Collary**. Suppose that $I$ is an interval. Let $f: I \rightarrow \mathbb{R}$ be continuous on $I$. Then, $f(I)$ is an intterval.
+**Corollary**. Suppose that $I$ is an interval. Let $f: I \rightarrow \mathbb{R}$ be continuous on $I$. Then, $f(I)$ is an intterval.
 
 **Theorem**. (Bolzano's) Invermediate Value Theorem. Suppose $f: [a, b] \rightarrow \mathbb{R}$ is continuous on $[a, b]$ with $a \neq b$. Then, given some $k$ such that $f(a) < k < f(b)$, there exists some $c \in (a, b)$ such that $k = f(c)$.
 
@@ -75,6 +75,6 @@ Note that if $f$ is uniformly continuous, it must be continuous on $A$.
 
 **Remark**. If $S, T$ are metric spaces, $K$ is a compact subset of $S$, and $f: K \rightarrow T$ is continuous on $K$, then $f$ is uniformly continuous.
 
-**Theorem**. Suppose $A \subseteq \mathbb{R}$ and $f: A \rightaarrow \mathbb{R}$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $A$, $(f(x_n))$ is a Cauchy sequence in $\mathbb{R}$.
+**Theorem**. Suppose $A \subseteq \mathbb{R}$ and $f: A \rightarrow \mathbb{R}$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $A$, $(f(x_n))$ is a Cauchy sequence in $\mathbb{R}$.
 
-**Remark**. Suppose $S, T$ are metric spaces and $f: S \rightaarrow T$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $S$, $(f(x_n))$ is a Cauchy sequence in $T$.
+**Remark**. Suppose $S, T$ are metric spaces and $f: S \rightarrow T$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $S$, $(f(x_n))$ is a Cauchy sequence in $T$.