## local-global principle for units

Say that $L/K$ is a quadratic extension of number fields with $K$ totally real and $L$ totally imaginary.

Then the Hasse norm theorem says that an element of $K$ that is everywhere a local norm is the global norm of something in $L$. (In fact, this is more generally true, whenever $L/K$ is cyclic.)

Is it also the case that an element of $\mathcal{O}_K^*$ that is everywhere a local norm is the global norm of something in $\mathcal{O}_L^*$?

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 In the question, do you mean a local norm of a unit, or just a local norm ? – François Brunault May 31 2011 at 12:29

No: there are units that are norms of elements but not norms of units. The simplest example are real quadratic number fields $Q(\sqrt{m}\,)$ with $m$ a sum of two squares such that the fundamental unit has positive norm, for example $m = 34$.
For finding other examples, one may look at the ambiguous class number formulas for cyclic extensions $L/K$ of prime degree $p$: $$Am(L/K) = h_K \frac{\prod e(P)}{p(E_K : E_K \cap NL^\times)}, \qquad Am_s(L/K) = h_K \frac{\prod e(P)}{p(E_K : N E_L)}.$$ Here $Am$ denotes the subgroup of ideal classes fixed by the Galois group $G$, $Am_s$ the subgroup of classes generated by ideals fixed by $G$, and $e(P)$ is the ramification index of the prime $P$. The index $$(Am:Am_s) = (E_K \cap NL^\times : NE_L)$$ is the obstruction to the local-global principle for units.
Edit. Let me, however, point out that D. Folk (When are global units norms of units?, Acta Arith. 76 (1996), 145-147) has proved the following result: if $L/K$ is normal and if $H$ denotes the Hilbert class field of $L$, then a unit from $K$ that is a local norm in all completions of $H/K$ is the norm of a unit from $L$. This suggests the following question: given a cyclic extension $L/K$, is there an unramified abelian extension $E/L$ with the property that a unit from $K$ is the norm of a unit from $L$ if and only if it is a local norm in all completions of $E/K$?