Exercise 5, § 17

Let $K_0$ be the field of rational fractions $\mathbf{Q}(X)$. Let K be the quadratic extension

$K_0(\rho )$, where $\rho$ is a primitive cubic root of unity, D be the cyclic extension $K(^{^3}X)$ of K, and $\sigma$ be a generator of the Galois group of D over K. Consider the algebra C defined as in Exercise 4, with $n= 3$ and $a= 2$.

a) Prove that C is a field with center K, of degree 9 over K. (To see that D has no element of norm 2, reduce to proving that there cannot be any relations of the form $2F^3= P^3$+XQ$^3+X^2R^3$ +3XPQR, where $F,P,Q,R$ are mutually prime polynomials in $\mathbf{Q}(\rho )[X]$. To do this, consider the lowest-degree terms of these polynomials.) b) Prove that there does not exist any automorphism of C extending the unique $K_0$-automorphism of K distinct from the identity (reduce to the case when such an automorphism preserves D). Deduce that $K_0$ is not a Galois subfield of C (VIII, p. 274, Exercise 16) and that there does not exist a $K_0$-automorphism of C of order 2.

c) Let $\xi$ be an element of D such that $\xi^3= X$, and let $\eta$ be an element of C such that $\eta^3= 2$ and $\eta \xi =\rho \xi \eta$; set $\zeta =\xi +\eta +\eta^2$. Prove that $K(\zeta )$ is a maximal commutative subfield of D and is not Galois over K, whereas the maximal commutative subfields $K(\xi )$ and $K(\eta )$ are Galois over K (calculate $\zeta^3$).