A latin square is an $n\times n$ array of $n$ symbols
in which each symbol appears exactly once in each row and column.
Regarding each symbol as a variable and taking the
determinant, we get a degree-$n$ polynomial in $n$ variables.
Can two latin squares $L,M$ have the same determinant,
up to a renaming of the variables, apart from the obvious cases
when $L$ is obtained from $M$ by a sequence of row
interchanges, column interchanges, renaming of variables, and transposition?
The answer was known to be no if $n\le7$; we show that it is yes for
$n=8$. The latin squares
for which this situation occurs have interesting special characteristics.