2 Let n K Q 1 If O has a system of n 1 fundamental units

2. Let n = [K : Q] > 1. If O? has a system of n ? 1 fundamental units, prove that K has no complex embeddings.

Solution

Let n = [K : Q] > 1. If O has a system of n 1 fundamental units, prove that K has no complex embeddings.

Recall Dirichlet\'s theorem on Units in a number field:

If K has r₁ real imbeddings and r₂ complex embeddings, then

              rank R of the unit group = r₁ +r₂-1.

In this case                      n-1= r₁ +r₂-1.

so                                         n= r₁ +r₂.......................(1)

On the other hand               n =r₁ +2 r₂......................(2)

(by the very definition of r₁ and r₂...

(2)-(1) gives                                 r₂=0.

In other words , the field K has no complex embeddings