The Principal Axis Theorem, included in most undergraduate texts in Linear Algebra though often without proof, states that every symmetric matrix over the field of real numbers is orthogonally similar to a diagonal matrix. In [1], S. Friedberg, focusing attention on the underlying field, gave an elementary argument to show that there are symmetric matrices over Zp (p a prime) which are not orthogonally similar to a diagonal matrix. This paper concludes with a problem: Classify exactly those fields for which the Principal Axis Theorem is true. As solutions to this classification problem can be found in the literature (see [2] and [10] for example) and frequent reconsiderations of this topic indicate an interest to a wide audience of mathematicians, the purpose of this paper is to give a simplified overview of this beautiful result. As we proceed, we keep an eye toward the accessibility of the argument. In fact, with the exception of two technical results, this development can be incorporated in any undergraduate-level course in Linear Algebra that deals with arbitrary fields. For example, one can show quite easily that it is necessary for the field to have characteristic equal to zero in order to insure that symmetric matrices are diagonalizable. While it is a rather straightforward matter to establish a large class of fields which allows for the orthogonal diagonalization of symmetric matrices, one of the aforementioned technical results is crucial in the final step of the classification of such fields. A field F is said to have the Principal Axis Property if every symmetric matrix over F is orthogonally similar to a diagonal matrix over F. That is, for every symmetric matrix M over F, there exists an orthogonal matrix P over F (that is, P= pt) such that P 1MP is diagonal. A study of the 2 x 2 case provides some important information. We write char(F) for the characteristic of F.