Exercise 3, § 3

Algebra, VII, p. 59

Let $(p_n)$ be a strictly increasing sequence of prime numbers such that $p_n$ does not divide any of the numbers

a + b(1 + p_1 + p_1 p_2 + \cdots + p_1 p_2 \cdots p_{n-1}) \quad \text{for} \quad |a| \leq n \quad \text{and} \quad |b| \leq n .

In the rational plane $\mathbf{Q}^2$ , consider the $\mathbf{Z}$ -module $M$ generated by the sequence $(x_i)$ defined as follows:

x_0 = (1, 0), \quad x_1 = (0, 1), \quad x_{n+1} = p_n^{-1}(x_0 + x_n) \quad \text{for } n \geq 1.

Show that $M$ has rank 2, and that every rank 1 submodule of $M$ is cyclic (notice that $x_0$ and $x_n$ form a basis for the submodule of $M$ generated by the $x_i$ with $i \leq n$ ).

Solution

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