Number fields with prescribed p-adic index

Let \(K=\mathbb {Q}(\theta )\) $K=\mathbb{Q}(\theta )$ be a number field generated by a root \(\theta \) $\theta$ of a monic irreducible polynomial \(g(x) \in \mathbb {Z}[x]\) $g(x)\in \mathbb{Z}[x]$ of degree n. We study the index of a number field, denoted by I(K), with a particular focus on its p-adic valuation \(\nu _p(I(K))\) ${\nu }_p(I(K))$ . We construct infinite families of number fields for which the p-adic valuation of the index is explicitly determined. In particular, we demonstrate that for both \(p=2\) $p=2$ and odd primes, the value of \(\nu _p(I(K))\) ${\nu }_p(I(K))$ can be arbitrarily large, irrespective of the degree n. Furthermore, we provide several examples to illustrate the results.