A new proof of the Pythagorean Theorem and generalization of the usual algebra norm

Abstract: A new proof of the Pythagorean Theorem is presented, utilizing George Birkhoff's version of the postulates of Euclidean geometry as incorporating the field of real numbers. Extrapolation of the mechanism of proof leads to a generalization of the usual algebra norm associated with a real finite-dimensional unital associative algebra, and resulting novel computationally accessible algebra isomorphism invariants.