Exercise 16, § 21

Suppose that the characteristic of K is zero. We say that an element $x$ of K is integral over $\mathbf{Z}$ if it belongs to a subring of K that is a finitely generated $\mathbf{Z}$-module or, equivalently, if the subring $\mathbf{Z}[x]$ of K is a finitely generated $\mathbf{Z}$-module $*$(cf.

Comm. Alg., V, §1, No. 1, p. $303)*$.

a) Prove that the set of elements of K integral over $\mathbf{Z}$ is a subring of K (for $x, y$ integral over $\mathbf{Z}$, consider the subring $\mathbf{Z}[x, y]$).

b) An element $x$ of K is integral over$\mathbf{Z}$ if and only if it is a root of a monic polynomial in $\mathbf{Z}[X]$ (when $x$ is integral, adapt the proof of Lemma 1 of VIII, p. 81 to show that it satisfies an equation of the form det($xI_m-A$) = 0 with $A\in \mathbf{M}_m(\mathbf{Z})$).

c) If $x$ is integral over $\mathbf{Z}$, then the same holds for its conjugates $x_1, . . . , x_m$ over $\mathbf{Q}$; the element $x_1\cdots x_m$ belongs to $\mathbf{Z}$ (use the lemma of VIII, p. 416).