Simultaneous Reduction of Quadratic Forms

Let x denote the real n-tuple (x1,…,xn), (xx… ) the iterated n-tuple (x1,…,xn,x1,…xn…), A = (a i j) and A\u27 a square matrix and its transpose. We wish to consider the effect of nonsingular transformations on a sum x(A1, + ... + Ak)x\u27 of k quadratic forms when the real symmetric matrices Ai and A = A, + ... + Ak are subjected to certain restrictions on their ranks. The results attained link a theorem in statistics with a theorem in ring theory (see Corollary 3). Extension of the results to hermitean forms follows easily

On high order derivations of fields

Let D ( L / K ) \mathcal {D}(L/K) denote the derivation algebra of a field extension L / K L/K of prime characteristic. If L / K L/K is purely inseparable and has an exponent, then every intermediate field F of L / K L/K equals the center of D ( L / F ) \mathcal {D}(L/F) . Here we prove the converse of this statement.</p

Separating 𝑝-bases and transcendental extension fields

Let L / K L/K denote an extension field of characteristic p ≠ 0 p \ne 0 . It is known that if L / K L/K has a finite separating transcendence base, then every relative p p -base of L / K L/K is a separating transcendence base of L / K L/K . In this paper we show that when every relative p p -base of L / K L/K is a separating transcendence base of L / K L/K , then the transcendence degree of L / K L/K is finite. We also illustrate the connection between the finiteness of transcendence degree of L / K L/K and the property that L / K ( X ) L/K(X) is separable algebraic for every relative p p -base X X of L / K L/K .</p