Exercise 17, § 2

Let $A$ be an integral domain. Put $n_k(A) = \dim(A[X_1, ..., X_k])$ for every integer $k \geqslant 0$. Let $\mathfrak{D}$ be the set of sequences of integers $(n_i)_{i \geqslant 0}$ such that there exists an integral domain $A$ with $n_i = n_i(A)$ for every integer $i \geqslant 0$. a) Prove the inequalities

b) Now prove that the only sequence of integers beginning with 1, 2 and belonging to $\mathfrak{D}$ is the sequence $n_i = i + 1$. (Let $A$ be such that $n_0(A) = 1$, $n_1(A) = 2$. By Exerc. 6, p. 84, the integral closure of $A$ is prüferian. Use Exerc. 6, $c$) , p. 84 to conclude.) c) Prove the inequalities

Deduce that if $n_0(A) = 0$, then $n_i(A) = i$ for every $i \in \mathbf{N}$. d) Prove that the sequence of differences $n_{i+1}(A) - n_i(A)$ is stationary. e) We use the notation of Exerc. 7, p. 84. Let $\delta$ be the transcendence degree of $L'$ over $L$. Then: (i) If $\delta = 0$, we have $n_i(A) = n_i(B) + 1$ for every $i$. (ii) If $1 \leqslant \delta < \infty$, we have $n_i(A) = n_i(B) + i + 1$ for $0 \leqslant i \leqslant \delta$ and $n_i(A) = n_i(B) + \delta + 1$ for $i \geqslant \delta$. (iii) If $\delta = \infty$, we have $n_i(A) = n_i(B) + i + 1$ for every $i$. (For a characterization of the elements of $\mathfrak{D}$, one may consult T. Parker, Amer. J. Math., 97 (1975), p. 308-311.)