Exercise 5, § 3

Let $k$ be a field and $K = k(X, Y)$ the field of rational functions in two indeterminates over $k$. We write $x_1 = X, y_1 = Y$ and for $n \geq 1$ define inductively the elements $x_n + , y_n$ and $y_n + , = x_n y_n^{-1}$ of K. (a) Write $A_n = k[x_n, y_n]$; the sequence $(A_n)$ is increasing and, if

$m_n$ is a maximal ideal of A, and $m_n = A_n \cap m_{n+1}$; in the ring $A = \bigcap_n A_n$, $m = \bigcap_n m_n$ is a maximal ideal. (b) Let $v$ be the valuation on K defined by $v(X) = \rho = (1 + \sqrt{5})/2$, $v(Y) = 1$; let B and p be the ring and ideal of $v$; show that $m = p \cap A$ (note that $\rho^2 - \rho - 1 = 0$ and use this fact to calculate $v(y_n)$). Show that for a monomial $X^i Y^j$ ($i, j$ positive or negative integers) to belong to A, it is necessary and sufficient that $\left( \frac{1 + \sqrt{5}}{2} \right)^i + j > 0$; for every monomial $z = X^i Y^j$, therefore, either $z \in A$ or $1/z \in A$. (c) Show that $B = A_m$ (write an element $t \in B$ in the form of a quotient of two polynomials in X and Y and show in each of them the monomials at which $v$ takes the least value). (d) Show that, for all $n$, $(A_{n+1})_{m_{n+1}}$ is the prime quadratic transform of $(A_n)_{m_n}$ in the direction of $v$ (Exercise 4).