Exercise 27, § 1

Let D be a field, $\sigma$ an automorphism of $D,d$ an endomorphism of the additive group of D satisfying relation (1) of VIII, p. 9, and A the ring $D[X]_{\sigma ,d}$ (VIII, p. 11). Given a nonzero element $P =\sum_na_nX^n$ of A, the greatest integer $n$ such that $a_n\not= 0$ is called the degree of P and denoted by deg(P). We set deg(0) $=-\infty$.

a) Let $F,G$ be elements of A with $G\not= 0$; show that there exist well-determined elements $Q,R$ (resp. $Q',R'$) of A such that F = QG + R and deg(R) $<$ deg(G) (resp. F = GQ$'+ R'$ and deg(R$'$)$<$ deg(G)). b) Deduce that every left (resp. right) ideal of A is monogenous. If $F,G,H,K$ are elements of A satisfying AF + AG = AH and AF $\cap$ AG = AK, then we have deg(F) + deg(G) = deg(H) + deg(K) (observe that the D-vector space $A/$AF has dimension deg(F)).

c) The ideal AF of A is maximal if and only if F is irreducible, that is, is not the product of two nonconstant polynomials.