Exercise 3, § 14

Let K be the field $\mathbf{C}(X)$ of rational fractions in one indeterminate over the (algebraically closed) field C of complex numbers. Show that in the ring $K[Y]$, the polynomial $Y^3 + X^3 + 1$ is irreducible ; let E be the extension of K generated by a root of this polynomial (in an algebraic closure of K). Show that E is not a pure extension of C, even though C is algebraically closed. (Show that it is impossible for a relation $u^3 + v^3 + w^3 = 0$ to exist between three polynomials u, v, w of $\mathbf{C}[X]$, pairwise relatively prime and not all three constant. Argue by contradiction : if r is the largest of the degrees of u, v, w and if for example $\deg(w) = r$, deduce from the relation

where j is a primitive cube root of unity, the existence of three polynomials $u_1, v_1, w_1$ in $\mathbf{C}[X]$, pairwise relatively prime, not all three constant, of degrees < r and such that $u_1^3 + v_1^3 + w_1^3 = 0.$ \*