Some Notes on the Tensor Product of Two Norm Forms

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l 1 and l 2 be cyclic field extensions of k of degree d with norms n l 1/k and n l 2/k . Minac's approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n l 1/k ⊗ n l 2/k of degree d. Let K = k(X 1,…, X d 2 ). We compute polynomials which are similarity factors of a form of the kind N ⊗ (n l 2/k ⊗ k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n l 1/k ⊗ n l 2/k to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n l 1/k .