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june4th notes complete
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To-Do Markdowns/LaTeX-notes/june4.log

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This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025) (preloaded format=pdflatex 2025.3.8) 4 JUN 2025 14:28
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This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025) (preloaded format=pdflatex 2025.3.8) 4 JUN 2025 14:48
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To-Do Markdowns/LaTeX-notes/june4.tex

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\newcommand{\Q}{\mathbb{Q}}
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\newcommand{\F}{\mathbb{F}}
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\newcommand{\K}{\mathbb{K}}
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\usepackage{fancyhdr}
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%}
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\definecolor{darkgreen}{rgb}{0.0, 0.5, 0.0}
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\definecolor{sasha}{rgb}{0.0, 0.6, 0.6}
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\definecolor{sasha}{rgb}{0.0, 0.5, 0.5}
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\definecolor{sam}{rgb}{0.2, 0.2, 0.8}
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\begin{document}
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\section*{June 4th—Example 1}
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\section*{June 4th}
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DEF Let $I$ be an ideal in $R$ (commutative ring).
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$$\sqrt I = \{r\mid r^n\in I\}$$
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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
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Thus, $J$ is an ideal.
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\end{proof}
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\color{black}
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\newpage
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\subsection*{Gandini Notes}{\newcommand{\ux}{\underline x}
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We'll start off with some preliminaries. \vpp
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DEF Let $A\in GL_n(\K).$\\
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We let $A$ "act" on $\underline x=(x_1,\cdots, x_n)$.
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by $A\cdot \ux=A\ux \leftarrow \text{MATRIX MULTIPLICATION}$. \vpp\color{red}
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DEF (Invariant Polynomial)\\
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$f(x_1,\cdots x_n)$ is invariant under $A$\\
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if $f(A\ux)=f(\ux)$ (so $f$ stays the same after ACTION)\color{blue}\vpp
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DEF Let $G$ be a group of marticies.
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If $f(A\ux)=f(\ux), \forall A\in G,$\\
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$f$ is an INVARIANT under $G$, on $f\in R^G$
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$$R^G=\{f\mid f\text{ is invariant under } G\}$$\vpp\color{sasha}
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SUBRING TEST S
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\begin{enumerate}
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\item A constant in $S$
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\item Closed under $+$
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\item Closed under $\cdot$
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\end{enumerate}
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\color{sam}
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\textbf{Sam's Work}:
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\begin{enumerate}
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\item $f(\ux)=C$\\
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$\rightarrow f(A\ux)=C$
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\item $f_1,f_@\in R^G$.\\
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WTS $f_1+f_@\in R^G$\\
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$f_2(A\ux)+f_2(A\ux)=f(\ux)+f_2(\ux)\in R^G$\\
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\item $f_1,f_2\in R^G$\\
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WTS $f_2f_2\in R^G$\\
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$f_1(A\ux)f_2(A\ux)$\\
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$=f_1(\ux)f_2(\ux)$ which is in $R^G$.
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\end{enumerate}\color{black}
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\subsubsection*{final gandini send-off}
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$\K[x^2]$ is the \color{red} subring generated by $x^2$\color{black}\\
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$(x^2)$ is the \color{red} ideal generated by $x^2$\\
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\color{sasha}$1\in $ SUBRING, \; $1\notin (x^2)$ as $(x^2)\neq R$. \\
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\color{darkgreen}$x^3\notin \K[x^2]$, \; $x^2\in (x^2)$
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}
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