Splitting Fields

Abstract

When we consider a polynomial such as X²+2 and extend the field ℚ to ℚ[i$\sqrt 2$] by adjoining one of the complex roots of the polynomial, we obtain a “bonus”, in that the other root −i$\sqrt 2$ is also in the extended field. Over ℚ[i$\sqrt 2$] we have that

$$ X^2 + 2 = (X - i\sqrt 2 )(X + i\sqrt 2 ), $$

We say that the polynomial splits completely (into linear factors) over ℚ[i$\sqrt 2$]. It is indeed clear that this must happen for a polynomial of degree 2, since the “other” factor must also be linear.