A positive Leopoldt defect is inherited by finite extensions

Setting and notation. Fix a natural number $p$ that is assumed prime (a typeclass hypothesis asserting the primality of $p$ is in force throughout). Let $F$ and $K$ be types, each equipped with a field structure and with the structure of a number field (a characteristic-zero field that is finite-dimensional over $\mathbb{Q}$). Assume in addition that $K$ is an $F$-algebra and that $K$ is finite-dimensional as an $F$-vector space; nothing further is assumed about the extension $K/F$ (no separability, normality, or nontriviality — the case $K = F$ is included, and no compatibility between the $F$-algebra structure on $K$ and the canonical $\mathbb{Q}$-structures on $F$ and $K$ is imposed beyond what the algebra structure itself gives).

For a number field $L$ (which will be $F$ and $K$ in turn), write $\mathcal{O}_L$ for its ring of integers, and define the following quantities, all of which are the unfolded content of the custom definitions used in the statement.

1. The primes above $p$. Let

$$S_p(L) \;=\; \{\, v \;:\; v \text{ a nonzero prime ideal of } \mathcal{O}_L \text{ with } p \cdot 1_{\mathcal{O}_L} \in v \,\}$$

i.e. the height-one primes of the Dedekind domain $\mathcal{O}_L$ that contain the image of the integer $p$. This set is finite (a finiteness instance is supplied, using that $p \ne 0$).

2. The semilocal unit group. For $v \in S_p(L)$ let $L_v$ be the completion of $L$ at the $v$-adic valuation and let $\mathcal{O}_{L_v} = \{x \in L_v : |x|_v \le 1\}$ be its valuation subring. Put

$$U_p(L) \;=\; \prod_{v \in S_p(L)} \mathcal{O}_{L_v}^{\times},$$

the (dependent) product of the unit groups of these local integer rings, with the pointwise group structure and the product topology, where each factor $\mathcal{O}_{L_v}^{\times}$ carries the topology induced by the embedding $u \mapsto (u, u^{-1})$ into $\mathcal{O}_{L_v} \times \mathcal{O}_{L_v}$.

3. The "unit closure" subgroup. Let

$$\Delta_L : \mathcal{O}_L^{\times} \longrightarrow U_p(L), \qquad u \longmapsto \big(\text{image of } u \text{ in } \mathcal{O}_{L_v}^{\times}\big)_{v \in S_p(L)}$$

be the diagonal homomorphism given componentwise by the structure map $\mathcal{O}_L \to \mathcal{O}_{L_v}$. Define

$$\mathcal{E}_p(L) \;=\; \bigcap_{n \ge 0} \Big\langle\, \operatorname{im}(\Delta_L) \;\cup\; \{\,x^{p^{\,n+1}} : x \in U_p(L)\,\} \,\Big\rangle,$$

the intersection, over all $n \in \mathbb{N}$, of the subgroup of $U_p(L)$ generated by the image of the global units together with the set of $p^{\,n+1}$-st powers of arbitrary semilocal units. Since $U_p(L)$ is commutative, each term of the intersection is the product subgroup $\operatorname{im}(\Delta_L) \cdot U_p(L)^{p^{\,n+1}}$, and the exponents range over $p, p^2, p^3, \dots$ (the exponent $p^0 = 1$ is not among them).

4. A bounded $\mathbb{Z}_p$-rank. Let $\mathbb{Z}_p$ denote the $p$-adic integers and, for $n \in \mathbb{N}$, regard $\mathbb{Z}_p^{\,n}$ as a group written multiplicatively, carrying the product $p$-adic topology. Define

$$\rho_p(L) \;=\; \sup\Big\{\, n \in \mathbb{N} \;:\; n \le [L : \mathbb{Q}] \ \text{ and }\ \exists\, f : \mathbb{Z}_p^{\,n} \to U_p(L) \text{ with } (\ast) \,\Big\},$$

where $(\ast)$ requires that $f$ be a homomorphism of groups (no $\mathbb{Z}_p$-linearity is demanded), that $f$ be injective, that $f$ be continuous (but not necessarily a topological embedding or a homeomorphism onto its image), and that $f(x) \in \mathcal{E}_p(L)$ for every $x \in \mathbb{Z}_p^{\,n}$. Here $[L:\mathbb{Q}]$ is the $\mathbb{Q}$-dimension of $L$. The supremum is taken in $\mathbb{N}$; the defining set always contains $n = 0$ (the trivial group maps injectively and continuously, with image $\{1\} \subseteq \mathcal{E}_p(L)$) and is bounded above by $[L:\mathbb{Q}]$, so $\rho_p(L)$ is the largest such $n$ and satisfies $0 \le \rho_p(L) \le [L:\mathbb{Q}]$.

5. The defect. Let $r(L) = \#\{\text{infinite places of } L\} - 1$ be the unit rank of $L$ in the sense of Dirichlet's unit theorem (the cardinality of the set of infinite places minus one, computed with truncated natural-number subtraction), and set

$$d_p(L) \;=\; r(L) \;\dot-\; \rho_p(L),$$

again using truncated subtraction on $\mathbb{N}$, so that $d_p(L) = 0$ whenever $\rho_p(L) \ge r(L)$, and $0 < d_p(L)$ holds exactly when $\rho_p(L) < r(L)$ strictly.

The assertion. With $p$, $F$, $K$ as above, the declaration asserts the implication

$$0 < d_p(F) \;\Longrightarrow\; 0 < d_p(K),$$

that is: if the unit rank of $F$ strictly exceeds $\rho_p(F)$, then the unit rank of $K$ strictly exceeds $\rho_p(K)$. The same prime $p$ is used on both sides. The hypothesis is a strict positivity of a natural number, i.e. the defect of $F$ is nonzero; the conclusion is the corresponding strict positivity for $K$. Because both quantities are differences taken with truncated subtraction, the statement carries no information about the sizes of the differences, only about whether each is nonzero.