\setlength{\parindent}{0pt}
\setlength{\parskip}{1em}

{
\renewtheorem{customlemma}{Lemma}[section]
\renewtheorem{customtheorem}[customlemma]{Theorem}
\renewtheorem{customproposition}[customlemma]{Proposition}
\renewtheorem{customcorollary}[customlemma]{Corollary}
\renewtheorem{customdefinition}[customlemma]{Definition}
\renewtheorem{customexample}[customlemma]{Example}
\renewtheorem{customremark}[customlemma]{Remark}
}

\newenvironment{examples}[1][(i)]{
\begin{example*}
\phantom{}
\begin{enumerate}[#1]
}
{
\end{enumerate}
\end{example*}
}

\newcommand\extendover{\glshyperlink[/]{field_ex_notation}}

\newcommand\degover[2]{\glshyperlink[\ensuremath{[#1 : #2]}]{ex_deg_notation}}

\newcommand\symmpoly{\glshyperlink[\ensuremath{s}]{symm_poly_notation}}

\newcommand\mult[1]{\glshyperlink[\ensuremath{#1^*}]{mult_group_notation}}

\newcommand\fieldadj[2]{\glshyperlink[\ensuremath{#1(#2)}]{field_adj_notation}}

\newcommand\ringadj[2]{\glshyperlink[\ensuremath{#1[#2]}]{field_adj_notation}}

\newcommand\Khom[1]{\glshyperlink[$#1$-homomorphism]{K_hom}}
\newcommand\Khompl[1]{\glshyperlink[$#1$-homomorphisms]{K_hom}}
\newcommand\Kisom[1]{\glshyperlink[$#1$-isomorphism]{K_hom}}
\newcommand\Kisomic[1]{\glshyperlink[$#1$-isomorphic]{K_hom}}
\newcommand\Kembedding[1]{\glshyperlink[$#1$-embedding]{K_hom}}
\newcommand\Kembeddingpl[1]{\glshyperlink[$#1$-embeddings]{K_hom}}

\def\fieldadjoin#1(#2){\fieldadj{#1}{#2}}
\let\fadj\fieldadjoin

\def\ringadjoin#1[#2]{\ringadj{#1}{#2}}
\let\radj\ringadjoin

\def\exdeg[#1 : #2]{\degover{#1}{#2}}

\newcommand\Disc{\glshyperlink[\ensuremath{\operatorname{Disc}}]{disc_f_notation}}

\newcommand\fd{\glshyperlink[\ensuremath{'}]{form_deriv_notation}}

\renewcommand\characteristic{\glshyperlink[\ensuremath{\operatorname{char}}]{char}}

\newcommand\fHom{\glshyperlink[\ensuremath{\Hom}]{homk_notation}}

\newcommand\algclos[1]{\ol{#1}}

\renewcommand\Aut{\glshyperlink[\ensuremath{\operatorname{Aut}}]{automorph_notation}}
\newcommand\KAut{\glshyperlink[\ensuremath{\operatorname{Aut}}]{Kautomorph_notation}}

\def\fixedf#1^#2{\glshyperlink[\ensuremath{#1^{#2}}]{fixed_field_notation}}

\newcommand\Gal{\glshyperlink[\ensuremath{\operatorname{Gal}}]{galois_group_notation}}

\newcommand\compfield[2]{\glshyperlink[#1#2]{comp_field_notation}}

\newcommand\aTrace{\glshyperlink[\ensuremath{\Trace}]{trace_alpha_notation}}
\newcommand\aNorm{\glshyperlink[\ensuremath{N}]{norm_alpha_notation}}

\newcommand\qFF{\glshyperlink[\ensuremath{\FF}]{finite_field_notation}}

\newcommand\Galpoly{\glshyperlink[\ensuremath{\operatorname{Gal}}]{galois_group_poly_notation}}
\newcommand\galover{\glshyperlink[\ensuremath{/}]{galois_group_poly_notation}}

\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\res}{res}

\newcommand\modred[1]{\glshyperlink[\ensuremath{\ol{#1}}]{modred_notation}}

\newcommand\runity{\glshyperlink[\ensuremath{\zeta}]{zeta_n_notation}}
\newcommand\runities{\glshyperlink[\ensuremath{\mu}]{mu_n_notation}}

\newcommand\cycpoly{\glshyperlink[\ensuremath{\Phi}]{cyc_poly_notation}}

\newcommand\cychom{\glshyperlink[\ensuremath{\chi}]{cyclotomic_hom_notation}}

\newcommand\multn[2][n]{\glshyperlink[\ensuremath{(#2^*)^{#1}}]{multn_notation}}

\newcommand\simeqto{\stackrel{\sim}{\to}}