M abstract-algebra-charles-pinter-notes.pdf → abstract-algebra-charles-pinter-notes.pdf

M abstract-algebra-charles-pinter-notes.tex → abstract-algebra-charles-pinter-notes.tex

@@ -104,10 +104,6 @@ \end{enumerate}
     we call $H$ a \emph{subgroup} of $G$. The operation of $H$ is the same as the operation of $G$.
 \end{definition}
 
-\begin{theorem}
-    Every subgroup of a cyclic group is cyclic.
-\end{theorem}
-
 \section{Functions}
 \begin{definition}[Function]
     If $A$ and $B$ are sets, then a function from $A$ to $B$ is a rule which to every element $x$ in $A$ assigns a unique element $y$ in $B$.

M galois-theory-notes.pdf → galois-theory-notes.pdf

M galois-theory-notes.tex → galois-theory-notes.tex

@@ -123,7 +123,7 @@ \begin{eg}{4.11}
   Beware, $L=\mathbb{Q}(i, -i, \sqrt{5}, -\sqrt{5}) = \mathbb{Q}(i, \sqrt{5}) = \mathbb{Q}(i+\sqrt{5})$.
 \end{eg}
 
-\begin{defn}{5.5}
+\begin{defn}{5.5}[Minimal polynomial]
   Let $L:K$, suppose $\alpha \in L$ is algebraic over $K$. Then, the \emph{minimal polynomial} of $\alpha$ over $K$ is the unique monic polynomial $m$ over $K$, $m(t) \in K[t]$, of smallest degree such that $m(\alpha)=0$.
   \\
   eg.: $i \in \mathbb{C}$ is algebraic over $\mathbb{R}$. The minimal polynomial of $i$ over $\mathbb{R}$ is $m(t)=t^2 +1$, so that $m(i)=0$.