  Constructing a polar code is all about selecting a subset of rows from a
Kronecker power of $[^1_1{}^0_1]$. It is known that, under successive
cancellation decoder, some rows are Pareto-better than the other. For instance,
whenever a user sees a substring $01$ in the binary expansion of a row index
and replaces it with $10$, the user obtains a row index that is always more
welcomed. We call this a "rule" and denote it by $10 \succcurlyeq 01$. In
present work, we first enumerate some rules over binary erasure channels such
as $1001 \succcurlyeq 0110$ and $10001 \succcurlyeq 01010$ and $10101
\succcurlyeq 01110$. We then summarize them using a "rule of rules": if $10a
\succcurlyeq 01b$ is a rule, where $a$ and $b$ are arbitrary binary strings,
then $100a \succcurlyeq 010b$ and $101a \succcurlyeq 011b$ are rules. This
work's main contribution is using field theory, Galois theory, and numerical
analysis to develop an algorithm that decides if a rule of rules is
mathematically sound. We apply the algorithm to enumerate some rules of rules.
Each rule of rule is capable of generating an infinite family of rules. For
instance, $10c01 \succcurlyeq 01c10$ for arbitrary binary string $c$ can be
generated. We found an application of $10c01 \succcurlyeq 01c10$ that is
related to integer partition and the dominance order therein.
