june4th notes complete

Gordie Novak · Gordie Novak · commit f241a7d39bee · 2025-06-04T14:48:30.000-05:00
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diff --git a/To-Do Markdowns/LaTeX-notes/june4.pdf b/To-Do Markdowns/LaTeX-notes/june4.pdf
diff --git a/To-Do Markdowns/LaTeX-notes/june4.tex b/To-Do Markdowns/LaTeX-notes/june4.tex
@@ -64,6 +64,7 @@
 \newcommand{\Q}{\mathbb{Q}}
 \newcommand{\F}{\mathbb{F}}
 \newcommand{\N}{\mathbb{N}}
+\newcommand{\K}{\mathbb{K}}
 \newcommand{\aut}[1]{{\rm Aut}(#1)}
 \newcommand{\Ker}{{\rm Ker}\,}
 \newcommand{\im}{{\rm Im}\,}
@@ -83,12 +84,13 @@
 \usepackage{fancyhdr}
 %}
 \definecolor{darkgreen}{rgb}{0.0, 0.5, 0.0}
-\definecolor{sasha}{rgb}{0.0, 0.6, 0.6}
+\definecolor{sasha}{rgb}{0.0, 0.5, 0.5}
 \definecolor{marcus}{rgb}{0.7, 0.3, 0.3}
+\definecolor{sam}{rgb}{0.2, 0.2, 0.8}
 
 
 \begin{document}
-\section*{June 4th—Example 1}
+\section*{June 4th}
 DEF Let $I$ be an ideal in $R$ (commutative ring).
 $$\sqrt I = \{r\mid r^n\in I\}$$
 \color{red}
@@ -128,6 +130,48 @@ \section*{June 4th—Example 1}
     So that means every term in the binomial sum is in $I$, and because $I$ is closed under addition, the sum must be in $I$, and therefore the sum must be in $J$. \vp
     Thus, $J$ is an ideal. 
 \end{proof}
+\color{black}
+\newpage
+\subsection*{Gandini Notes}{\newcommand{\ux}{\underline x}
+
+We'll start off with some preliminaries. \vpp
+DEF Let $A\in GL_n(\K).$\\
+We let $A$ "act" on $\underline x=(x_1,\cdots, x_n)$.
+by $A\cdot \ux=A\ux \leftarrow \text{MATRIX MULTIPLICATION}$. \vpp\color{red}
+DEF (Invariant Polynomial)\\
+$f(x_1,\cdots x_n)$ is invariant under $A$\\
+if $f(A\ux)=f(\ux)$ (so $f$ stays the same after ACTION)\color{blue}\vpp
+DEF Let $G$ be a group of marticies.
+If $f(A\ux)=f(\ux), \forall A\in G,$\\
+$f$ is an INVARIANT under $G$, on $f\in R^G$
+$$R^G=\{f\mid f\text{ is invariant under } G\}$$\vpp\color{sasha}
+SUBRING TEST S 
+\begin{enumerate}
+    \item A constant in $S$
+    \item Closed under $+$
+    \item Closed under $\cdot$
+\end{enumerate}
+\color{sam}
+\textbf{Sam's Work}:
+\begin{enumerate}
+    \item $f(\ux)=C$\\
+        $\rightarrow f(A\ux)=C$
+    \item $f_1,f_@\in R^G$.\\
+        WTS $f_1+f_@\in R^G$\\
+        $f_2(A\ux)+f_2(A\ux)=f(\ux)+f_2(\ux)\in R^G$\\
+    \item $f_1,f_2\in R^G$\\
+        WTS $f_2f_2\in R^G$\\
+        $f_1(A\ux)f_2(A\ux)$\\
+        $=f_1(\ux)f_2(\ux)$ which is in $R^G$. 
+\end{enumerate}\color{black}
+\subsubsection*{final gandini send-off}
+$\K[x^2]$ is the \color{red} subring generated by $x^2$\color{black}\\
+$(x^2)$ is the \color{red} ideal generated by $x^2$\\
+\color{sasha}$1\in $ SUBRING, \; $1\notin (x^2)$ as $(x^2)\neq R$. \\
+\color{darkgreen}$x^3\notin \K[x^2]$, \; $x^2\in (x^2)$
+}
+
+
 
 
 
