# Prime field of a skew field is a field

I'm having a question about skew fields (or division rings). The intersection of all the non-trivial sub skew fields is called the prime field. Now it says that this prime field is a field because it is a subfield of the center of the skew field. (The center is defined by all the elements of the skew field who commute with all the other elements of the skew field.) But what happens when the center is trivial? How can you prove that the prime field is really a field?

Ofcourse, when the skew field is finite, this question is trivial because of Wedderburn's theorem.

The centre isn't trivial. It clearly contains $1$.