My friend has written an introduction to algebraic number theory before, which contains a short proof of this statement, but I didn't check its validity.
http://ihome.ust.hk/~masoarer/notes/intro_ant.pdf
[Edit: Update the link of the document] http://www.2shared.com/document/2d6M7kNU/Introduction_to_Algebraic_Numb.html p.41 of the document, or p.45 of the pdf.

