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 $\mathcal{O}_L^*$?
 A: 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 $E/K$?
