Exercise 12, Appendix 1

a) Suppose that $k$ is a field, that the $k$-algebra A is Artinian, and that every increasing sequence of subalgebras of A is stationary. Prove that A has finite degree over $k$. (Reason as in Exercise 11. Note that the field $k(X)$ contains an infinite strictly increasing sequence of subalgebras, using VII, §1, No. 5, p. 6, Proposition 6.) b) Prove that in the polynomial algebra $k[X]$, every increasing sequence of subalgebras is stationary. (Let $f\in k[X]$ be a nonconstant polynomial and A be the $k$-subalgebra of $k[X]$ generated by 1 and $f$. Observe that A is a principal ideal domain and $k[X]$ is a finitely generated A-module.)