Exercise 2, § 14

Let K be the field $\mathbf{Q}(X)$ of rational fractions in one indeterminate over the field $\mathbf{Q}$ of rational numbers. Show that in the ring $K[Y]$ the polynomial $Y^2 + X^2 + 1$ is irreducible and if E is the extension of K generated by a root of this polynomial (in an algebraic closure of

K), then Q is relatively algebraically closed in E but E is not a pure extension of Q. (To see that there is in E no element a $\notin$ Q algebraic over Q note that the existence of such an element would imply the relation $E = \mathbf{Q}(a)(X)$, and observe that $- (X^2 + 1)$ is not a square in $\Omega(X)$, where $\Omega$ is an algebraic closure of Q.) Show that if i is a root of $X^2 + 1$, then $E(i)$ is a pure transcendental extension of $\mathbf{Q}(i)$.