ISOMORPHISM OF QF ALGEBRAS OVER ℚ

We show that the local–global principle (Hasse principle) holds for isomorphisms between commutative local QF algebras over ℚ of dimension five with radical cubed zero, and obtain a criterion for them to be isomorphic by the Hilbert symbol or the Legendre symbol. As a consequence, we give the isomorphism classes of some type of local algebras over ℚ by using a factor group of ℚ*.