Division Algebras of Higher Degree over Rational Function Fields in One Variable

Let k be a field and ℓ a prime not equal to the characteristic of k. Assume that k contains a primitive ℓ^th ischolar_main of unity. Given a central simple algebra D over k(T) with ramification index at most 3, we show that there exist a centrals simple algebra D0 over k and f, g ∈ k(T)* such that D is Brauer equivalent to the tensor product of D₀ and the cyclic algebra (f, g)ℓ over k(T). We also give necessary and sufficient conditions for the algebra D₀⊗ (f, g)ℓ to be division for some class of polynomials f and g.