Proposition 9

— If K is a maximal ordered field, then the only irreducible polynomials in K[X] are the first degree polynomials, and the second degree polynomials $aX^2 + bX + c$ such that $b^2 - 4ac < 0$.

Since $K(i)$ is algebraically closed, every algebraic extension of K, and hence also every irreducible polynomial over K, has degree 1 or 2. To see which second degree polynomials are irreducible, it is enough to consider the canonical form $a((X + (b/2a))^2 - (b^2 - 4ac)/4a^2)$ (cf. VI, p. 26, Prop. 7).