The equation $x^2-x-1$ has no solution over finite fields of even order.

Question

Is the equation $x^2-x-1$ has no solution over $GF(2^i)$ for all i. I can prove it trivially for some arbitrary chosen small even ordered fields, but can this be generalized?

Answer 1

Note that in any characteristic $2$ field, $-1 = 1$ and the polynomial is equal to $x^2 + x + 1$. This has a root over the field of four elements.

We can think of that field very explicitly; it is given by $0, 1, \alpha, \beta$, with $\alpha = 1 + \beta$ and the rest of the addition table following from the fact that $1 + 1 = 0$ in characteristic $2$.

The multiplication rules are $\alpha^2 = \beta$, $\alpha\beta = 1$, $\beta^2 = \alpha$, and again the rest of the table follows from standard rules for rings.

You can check that both $\alpha$ and $\beta$ are roots of your polynomial.