Argument of begin has an extra } when trying to use cases

I’ve just moved to compiling LaTeX on my desktop instead of using Overleaf and all’s going fine, but I can’t seem to be able to use the cases environment. Here’s my code (some lines removed for brevity):

documentclass[12pt]{report}

usepackage{polski}

usepackage{amsmath,amsfonts,amssymb}
usepackage{tikz-cd}

%wieksze marginesy
usepackage[margin=1in]{geometry}

begin{document}

%(...............)

Oczywiście $(1, ldots, 1) in W^perp$. Gdy $mathrm{char}kmid n$, to również $(1, ldots, 1) in W$, a zatem $W cup W^perp neq {0}$ i $W, W^perp$ są podprzestrzeniami zdegenerowanymi. Niech $mathrm{char}k nmid n$. Załóżmy, że $(a_1, ldots, a_n) notin mathrm{lin}{(1, ldots, 1)}, (a_1, ldots, a_n) in W^perp$. Istnieją $i, j in {1, ldots, n}$ takie, że $a_i neq 0$, $a_j neq a_i$. Załóżmy dla wygody, że $i = 1$, $j = 2$. Rozważmy układ równań

$$
begin{cases}
    x_1 + ldots + x_n & = 0 
    a_1x_1 + ldots + a_nx_n & = 0
end{cases}
xrightarrow
begin{cases}
    x_1 + ldots + x_n & = 0 
    (frac{a_2}{a_1} - 1)x_2 + ldots + (frac{a_n}{a_1} - 1)x_n & = frac{1}{a_1}
end{cases}
$$

end{document}

Compiling with pdflatex gives

Underfull hbox (badness 10000) in paragraph at lines 26--28

! Argument of begin  has an extra }.
<inserted text> 
                par 
l.37 begin
           {cases}
? 
Runaway argument?
! Paragraph ended before begin  was complete.
<to be read again> 
                   par 
l.37 begin
           {cases}
? 
! Missing $ inserted.
<inserted text> 
                $
l.37 begin
           {cases}
? 
! Missing } inserted.
<inserted text> 
                }
l.37 begin
           {cases}
? 
! Missing $ inserted.
<inserted text> 
                $
l.37 begin
           {cases}
? 
! Limit controls must follow a math operator.
ext@arrow ...x towd tw@ {unhbox z@ }}limits 
                                                  @ifnotempty {#7}{^{if 0#...
l.37 begin
           {cases}
? 
! Missing endcsname inserted.
<to be read again> 
                   def 
l.37 begin
           {cases}
? 
)
! Incomplete ifx; all text was ignored after line 37.
<inserted text> 
                fi 
<*> birula.tex

I’ve tried a couple of things but to me, my use of cases just seems completely inoffensive. What could be the problem?