Skip to main content\( \renewcommand{\le}{\leqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\ge}{\geqslant}
\newcommand{\ideal}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\subgp}[1]{\left\langle\, #1 \,\right\rangle}
\def\p{\varphi}
\def\isomorphic{\cong}
\def\Gal{\text{Gal}}
\DeclareMathOperator{\im}{im}
\def\normal{\vartriangleleft}
\def\lcm{{\text{lcm}\,}}
\newcommand{\q}[1]{{\color{red} {#1}}}
\newcommand{\set}[1]{\left\{ {#1} \right\}}
\newcommand{\setof}[2]{{\left\{#1\,\colon\,#2\right\}}}
\def\C{{\mathbb C}}
\def\Z{{\mathbb Z}}
\def\bF{{\mathbb F}}
\def\Q{{\mathbb Q}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\newcommand{\h}[1]{{\textbf{#1}}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
