On the equation \(x^{2^l+1}+x+a=0\) over \(\mathrm{GF}(2^k)\) (Q2469479)

The authors look at polynomials \(f(x)\) of degree \(2^\ell+1\) over \(\mathbb F= \text{GF}(2^k)\) and determine criteria for the number of zeros of \(f(x)\) over \(\mathbb F\). In particular, when \(\gcd(\ell,k)=1\), they present conditions for \(f(x)\) to have exactly one zero over \(\mathbb F\), and actually calculate this root explicitly.