Skew Laurent extensions of domains are domains ($\mathrm{lc}(fg)=a_m{\sigma }^m(b_n)$)

Let $A$ have no zero divisors and $\sigma$ be an automorphism. For nonzero Laurent polynomials $f=\sum a_i{x}^i$ and $g=\sum b_j{x}^j$ with greatest exponents $m,n$, the coefficient of $x^{m+n}$ in $fg$ is $a_m{\sigma }^m(b_n)$. No other exponent pair contributes to that highest degree. Both factors are nonzero, so their product is nonzero, proving the noncommutative domain property in every characteristic.