2024 Wrapup

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

@@ -50,3 +50,33 @@ Additionally, if $R$ is a PID, as $\Ann_R(N)$ is an ideal, $\Ann(N) = (n)R$ and
 
 1. $N$ is a free submodule with rank $n \leq m$.
 2. There exists a basis $y_1, y_2, \ldots, y_m$ of $M$ so that $r_1 y_1, r_2 y_2, \ldots, r_m y_n$ is a basis of $N$ for some $r_i \in R$ and $r_1 | r_2 | \ldots | r_n$
+
+**Theorem**. Fundamental Theorem, Existence: Invariant Form. Let $R$ be a PID and $M$ be a finitely generated $R$-module. THen,
+
+- $M$ is isomorphic for some $r \in \mathbb{N}\cup{0}$, $a_1, \ldots, \a_m \neq 0 \in R$ such that $a_1 | a_2 | \ldots | a_m$, with
+
+$$
+M \cong R^{\oplus r} \oplus \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}
+$$
+
+- $M$ is torsion-free if and only if $M$ is free
+
+- Note that
+
+$$
+\Tor{M} \cong \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}
+$$
+
+As a consequence, $M$ is a torsion module if and only if $r = 0$.
+
+**Definition**. In the above, $r$ is the *free rank* of $M$, and $a_1, \ldots, a_m$ are the *invariant factors* of $M$.
+
+**Theorem**. Fundamental Theorem, Existence: Elementary Divisor Form. The sum above can be written as
+
+$$
+M \cong R^{\oplus r} \oplus \frac{R}{(p_1^{\alpha_1})R} \oplus \frac{R}{(p_2^{\alpha_2})R} \oplus \ldots \oplus \frac{R}{(p_t^{\alpha_t})R}
+$$
+
+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

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

@@ -0,0 +1,19 @@
+# Dummit & Foote Chapter 12 - 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**. If $K, F$ are fields such that $F \subseteq K$, then $K$ is an *extension field* or *extension* of $F$, denoted $K / F$.
+
+**Definition**. The *degree* (or *relative degree* or *index*) of $K/F$, denoted $[K:F]$, is the dimension of $K$ as a $F$-vector space.
+
+**Theorem**. Let $F$ be a field, $p(x) \in F[x]$. Then, there exists a $K$ such that $p(x)$ has a root in $K$.
+
+**Theorem**. Let $F$ be a field, $p(x) \in F[x]$. Then, $K = \frac{F[x]}{(p(x))}$ and $\theta = x \amod{p(x)}$, $K$ has a basis of $1, \theta, \ldots, \theta^{n-1}$ where $n = \deg(p)$.
+
+**Theorem**. Let $K/F$ and $\alpha, \beta, \ldots \in K$. Then, the smallest subfield of $K$ containing $F$ and $\alpha, \beta, \ldots$ is $F(\alpha, \beta, \ldots)$, which is the *field generated by $\alpha, \beta, \ldots$ over $F$*.
+
+**Definition**. If $K$ is generated by $F(\alpha)$, then $K$ is a *simple extension* of $F$.
+
+**Theorem**. Let $F$ be a field, $p(x) \in F[x]$ be irreducible. Then, if $\alpha$ is a root of $p(x)$ and $K$ is an extension of $F$ containing $\alpha$, then $F(\alpha) \cong \frac{F[x]}{(p(x))}$.