All the Codeword Symbols in Polar Codes Have the Same SER Under the SC Decoder

Let
${\mathbb F}_{p}$
be a prime field and let
$\mathbb {F}_{q}$
be a larger finite field obtained from adjoining an element
$\alpha $
to
${\mathbb F}_{p}$
, i.e.,
$\mathbb {F}_{q}= {\mathbb F}_{p}(\alpha)$
. We consider polar codes constructed from the
$2\times 2$
kernel
$\begin{aligned} \begin{bmatrix} 1 & 0 \\ \alpha & 1 \end{bmatrix} \end{aligned}$
over
$\mathbb {F}_{q}$
. We prove that for any
$\mathbb {F}_{q}$
-symmetric memoryless channel, any code length, and any code dimension, all the codeword symbols in such polar codes have the same symbol error rate (SER) under the successive cancellation (SC) decoder.