Exercise 4, § 17

Let D be a field, $\sigma$ an automorphism of D, A the ring $D[X]_{\sigma}$ (VIII, p. 11)$,n$ an integer $\geqslant 2$, and $a$ a nonzero element of D. Denote the element $X^n-a$ of A by $f$. a) The ideal $Af$ is two-sided if and only if we have $\sigma (a) =a$ and $\sigma^n(x) =axa^{-1}$ for every $x\in D$. b) Suppose that the group $\mu_n$ of $n$-th roots of unity contained in the center of D is cyclic of order $n$ and fixed by $\sigma$. Let $g\in A$ be an irreducible monic polynomial (VIII, p. 22, Exercise 27) dividing $f$ on the right. Prove that the degree $d$ of $g$ divides $n$ and that $f$ is the product of $n/d$ polynomials of degree $d$. (Let $h$ be the monic polynomial such that $Ah=\bigcap_{\zeta\in\mu_n}Ag_{\zeta}$, where $g_{\zeta}(X) =g(\zeta X)$. Observe that we necessarily have $h(\zeta X) =h(X)$ for every $\zeta \in \mu_n$, and deduce that $h=f$. Then, apply loc. cit., b) by constructing, for $1\leqslant i\leqslant n/d$, an element $\zeta_i$ of $\mu_n$ and a polynomial $h_i$ of degree $id$ such that $Ag_{\zeta_1}\cap \cdots \cap Ag_{\zeta_i}= Ah_i.)$

If, moreover, the ideal $Af$ is two-sided, then the ring $A/Af$ is semisimple, and its simple components all have the same length (observe that every simple $(A/Af$)-module is isomorphic to one of the modules $A/Ag_{\zeta})$.

c) Assume from now on that the hypotheses of b) are satisfied, that the ideal $Af$ is two-sided, and that $n$ is a prime number. Deduce from b) that we are in one of the following situations:

(i) There exists a $b\in D$ such that $a=b^n$ and that $\sigma$ is the inner automorphism associated with $b$. The ring $C = A/Af$ is isomorphic to the product ring $D^n$.

(ii) The condition of (i) is not satisfied, but there exists a $c\in D$ such that $a=\sigma^{n-1}(c)\sigma^{n-2}(c). . . \sigma (c)c$. The ring C is simple of length $n$.

(iii) There is no element $c$ of D such that $a=\sigma^{n-1}(c)\sigma^{n-2}(c)\cdots \sigma (c)c$. The ring C is a field.

In the case when D is commutative, recover the results of Exercise 11 of VIII, p. 330. If, moreover, $n= 2$, then C is a quaternion algebra over the subfield K of D invariant under $\sigma$.

d) Prove that there exists an automorphism $\tau$ of C of order $n$ whose subfield of invariants is equal to D. In cases (ii) and (iii) above, D is a Galois subfield of C (Exercise 16 of VIII, p. 274).

e) Suppose that $\sigma$ is the inner automorphism associated with an element $d$ of D. Prove that we have $a=d^nz$, where $z$ belongs to the center Z of C, and that the ring C is isomorphic to the tensor product $D\otimes_ZZ[X]/(X^n-z)$.

f ) Suppose that $\sigma$ is not an inner automorphism. Prove that the center of C is the subfield of Z consisting of the elements invariant under $\sigma$.