style: clear trailing whitespace

flake.nix

@@ -14,8 +14,8 @@
           inherit (pkgs.texlive)
           scheme-medium
           asymptote wrapfig amsmath ulem hyperref capt-of
-          latexmk biber xpatch 
-          tkz-graph tikz-cd xcolor todonotes 
+          latexmk biber xpatch
+          tkz-graph tikz-cd xcolor todonotes
           mdframed mathtools braket
           multirow prerex cleveref
           wasysym stmaryrd

patch-asy.sty

@@ -7,15 +7,15 @@
 %% asy-latex.dtx  (with options: `pkg')
 %% ____________________________
 %% The ASYMPTOTE package
-%% 
+%%
 %% (C) 2003 Tom Prince
 %% (C) 2003-2010 John Bowman
 %% (C) 2010 Will Robertson
-%% 
+%%
 %% Adapted from comment.sty
-%% 
+%%
 %% Licence: GPL2+
-%% 
+%%
 \ProvidesPackage{asymptote}
   [2012/08/25 v1.27 Asymptote style file for LaTeX]
 \def\Asymptote{{\tt Asymptote}}

references.bib

@@ -148,7 +148,6 @@ @unpublished{ref:gorin
 	year={2018},
 	url={https://www.mit.edu/~txz/links.html},
 }
-	
 
 // Notes used in passing
 @unpublished{ref:covering_all_we_know,

tex/alg-NT/frobenius.tex

@@ -600,7 +600,7 @@ \section{Frobenius elements control factorization}
 
 To do this, suppose $S = \{ \alpha_1,\alpha_2,\dots, \alpha_n\}$ are the roots of $f$ (distinct roots since $f$ is irreducible over $\QQ$). We let $\Frob_{\kP}$ act on $S$. This splits $S$ into orbits $S_1$, $S_2$, $\dots$, $S_k$. Construct polynomials $f_i$ with coefficients in $E$ having roots exactly the elements of $S_i$. This forms a factorization of $f$ over $E$, say \[ f = f_1f_2 \dots f_k. \]
 
-We claim that this in fact induces a factorization of $f \pmod p$. 
+We claim that this in fact induces a factorization of $f \pmod p$.
 To see this, consider the images of these polynomials $f_i$ under the quotient $\OO_K \to \OO_K/\kP$, denote them by $\overline{f_i}$. Then since $p$ is unramified, we know that the decomposition group $D(\kP|p)$ is isomorphic to the Galois group $\mathcal{G} = \Gal((\OO_E/\kP) / (\ZZ/p\ZZ))$. Thus $\Frob_{\kP}$ corresponds to the generator $\sigma$ of $\mathcal{G}$. It is not hard to believe that the action of $\Frob_{\kP}$ on the roots of $f$ is the same as that of $\sigma$ on the roots of $ \overline{f}$. Since the roots of $f_i$ form an orbit under the action of $\Frob_{\kP}$, we see that the roots of $\overline{f_i}$ form an orbit under the action of $\sigma$ and hence under the action of $\mathcal{G}$. It is now a standard fact of Galois theory that $\overline{f_i}$ is an irreducible polynomial over $\FF_p$ (since it is fixed by $\mathcal{G}$), thus the claim is proved.
 
 Now we just need to observe that the roots of $f$ correspond to the cosets of $H$, this will be established later.

tex/frontmatter/title-embellishments.tex

@@ -19,7 +19,7 @@
 		\url{https://ko-fi.com/evanchen/}
 	\end{minipage}
 	\end{center}
-	
+
 	\vfill
 	{
 	\small

tex/measure/martingale.tex

@@ -580,7 +580,7 @@ \section{\problemhead}
 	Finally let $\tau \colon \Omega \to \{1, 2, \dots\}$
 	be a stopping time such that $\mathbb E[\tau] < \infty$,
 	such that the event $\tau = n$ depends only on $X_1$, \dots, $X_n$.
-	
+
 	Prove that
 	\[ \EE[X_1 + X_2 + \dots + X_\tau] = \mu \EE[\tau]. \]
 \end{problem}
@@ -621,4 +621,3 @@ \section{\problemhead}
 		(Incidentally, using the law of large numbers could work too.)
 	\end{hint}
 \end{problem}
-

tex/rep-theory/characters.tex

@@ -494,7 +494,7 @@ \section{Examples of character tables}
 	Here, $C_G(g) = \left\{ x \in G : xg = gx \right\}$
 	is the centralizer of $g$.
 	\begin{hint}
-		Construct two square $r \times r$ matrices $A$ and $B$ 
+		Construct two square $r \times r$ matrices $A$ and $B$
 		such that $AB$ is the identity by the first orthogonality.
 		Then use $BA$ to prove the second orthogonaliy relation.
 	\end{hint}