Interpreting finite fields in towers of cyclotomic fields(Model theoretic aspects of the notion of independence and dimension)
Kenji Fukuzaki · Institutional Repositories DataBase (IRDB) · 2007
Let $l$ be an odd prime and $\zeta_{l^{n}}$ is a primitive $l^{\mathfrak{n}}$ -th root of unity.We consider the towers of cylotomic fields $K_{l}= \bigcup_{n}\mathbb{Q}(\zeta_{l^{n}})$ .We prove that, for any positive integer $k$ , there is a prime $p>k$ such that $\mathbb{Z}/(p)$ is interpretable in $K_{l}$ .The proof uses the method of Julia Robinson by which she proved the undecidability of number fields.For $K_{m}= \bigcup_{n}\mathbb{Q}(\zeta_{m^{n}})$ , where $m$ is $an$ arbitrary positive integer and $\zeta_{m^{n}}$ is a primitive $m^{n}$ -th root of unity, we prove that for any positive integer $k$ , there is a prime $p>k$ such that some finite product of $\mathbb{Z}/(p)$ is interpretable in $K_{m}$ . Lemma 1 $h\in F^{*}$ can be represented by the form $x^{2}-ay^{2}-bz^{2}iff-ab/h ot\in F_{\mathfrak{p}^{r2}}$ for any valuation $\mathfrak{p}such$ that $(a, b)_{\mathfrak{p}}=-1$ .This follows the property of quaternary quadratic forms and the Hasse-Minkowski theorem on quadratic forms.See [4, p. 187] and [6, p.lll].Using this lemma, J. Robinson proved the following:( \dagger ) Let $m$ be a positive integer such that $\mathfrak{p}^{m}\parallel 2$ for all prime ideds $\mathfrak{p}$ .Let $\varphi(s, u,t)$ $be$ $\exists x,$ $y,$ $z(1-sut^{2m}=x^{2}-sy^{2}-uz^{2})$ .For $t ot\in O$ , there are $a,$ $b\in D$ such that 1. $F\models eg\varphi(a, b,t)$ , 2. $F\models\forall c(\varphi(a, b, c)arrow\varphi(a, b, c+1))$ .