sage: K = QQ[2^(1/3)]; K Number Field in a with defining polynomial x^3 - 2 sage: K.complex_embeddings() [Ring morphism: ... Defn: a |--> -0.629960524947 - 1.09112363597*I, Ring morphism: ... Defn: a |--> -0.629960524947 + 1.09112363597*I, Ring morphism: ... Defn: a |--> 1.25992104989]
Let be the map , and let be the -span of the image of inside .
Next assume that is discrete and let be any positive number. Then for every there is an open ball that contains but no other element of . Since is closed and bounded, it is compact, so the open covering of has a finite subcover, which implies that is finite, as claimed.
Since is discrete in , Lemma 6.1.3 implies that equals the rank of . Since is injective, is the rank of , which equals by Proposition 2.4.5.