The authors are concerned with finding conditions under which a central division algebra \(D\) of prime degree \(p\) over its centre \(F\) is cyclic. More generally let \(D/F\) be a central division algebra of degree \(n\) and assume that \(D\) contains a maximal subfield \(K\) such that for a cyclic field extension \(E/K\) of degree \(q\) prime to \(n\), the extension \(E/F\) is Galois with a semidirect product \(C_n\rtimes C_q\) as Galois group. An induction reduces this situation to the case where \(n=p^m\) is a prime power. Let \(\sigma\) be a generator of \(C_n\) and \(L=E^\sigma\) the fixed field. Further let \(B\) be the group ring of \(C_q\) over the integers \(\text{mod }n\). The group \(A=L^\times/(L^\times)^n\) may be regarded as a \(B\)-module. For any \(a,b\in A\) denote by \((a,b)_n\) the symbol algebra [of \textit{L. H. Rowen}, Ring Theory II (Academic Press 1988; Zbl 0651.16002), p. 194] and consider the mapping \(A\times A\to\text{Br}(L)\) which takes the pair \(a\), \(b\) to the class of \((a,b)_n\). This mapping is skew-symmetric and so defines a mapping \(\phi\colon A\wedge A\to\text{Br}(L)\), which is actually a \(B\)-homomorphism. For the algebra \(D\) as above, \(D_L\) is shown to be an image of a decomposable element \(a\wedge b\) of \(A\wedge A\). More generally, given any central simple algebra \(D/F\) of degree \(n\), \(D\) is called \(q\)-accessible if there is a \(C_q\)-Galois extension \(L/F\) such that the Brauer class of \(D_L\) is the image under \(\phi\) of a \(C_q\)-invariant decomposable element \(\alpha\) of \(A\wedge A\). Such an \(\alpha\) defines a \(B\)-submodule \(M\) of \(A\) of rank 2, such that \(M\wedge M\cong (Z/nZ)\alpha\). Conversely a \(B\)-submodule \(M\) of \(A\) of rank 2, such that \(M\wedge M\) has trivial \(C_q\)-action, corresponds to a decomposable \(C_q\)-fixed element of \(A\wedge A\); such \(M\) is called accessible. The module \(M\) associated to a \(q\)-accessible algebra \(D\) is not unique; therefore \(D\) is also described as \((q,M)\)-accessible. Conversely given a \(C_q\)-Galois extension \(L/F\) and an accessible \(B\)-submodule \(M\) of \(A\), with generator \(\alpha\), say, then there is a \((q,M)\)-accessible \(D/F\) of degree \(n\) such that the class of \(D_L\) is the image \(\alpha\phi\). Here \(D\) is unique up to choice of \(\alpha\), and changing \(\alpha\) changes \(D\) by a power prime to \(n\) in \(\text{Br}(F)\). The authors' aim now is to construct a generic \((q,M)\)-accessible algebra. Given an accessible \(B\)-module \(M\) and a surjective \(C_q\)-morphism \(P\to M\), where \(P\) is a finitely generated \(Z\)-free \(Z[C_q]\)-module, we can form the group algebra \(F[P]\) and its field of fractions \(F(P)\). Denote by \(T\) and \(K\) the fixed ring resp. field under the action of \(C_q\). Then there is a division algebra \(D/K\) of degree \(n\) which is generic for \((q,M)\)-accessible algebras containing \(F\) in their centre. In particular if \(D/K\) is cyclic, then any \((q,M)\)-accessible algebra containing \(F\) in its centre is cyclic. If \(M\) is an accessible \(C_q\)-module which is faithful (i.e. no non-trivial subgroup acts trivially), then the least rank of a \(C_q\)-lattice \(P\) mapping onto \(M\) is \(\max(2,\varphi(q))\) (\(\varphi=\)Euler function). Thus \(P\) has rank 2 only for \(q=2,3,4,6\). These results are illustrated by proving (i) any 2-accessible algebra \(D/K\) of degree \(n\), where \(K\) has characteristic 0 and contains a primitive \(n\)-th root of 1, is cyclic; the same holds for a 4-accessible algebra. (ii) By computations using Mathematica and results of \textit{T. Ford} [New York J. Math. 1, 178-183 (1995)], the authors show that any 3-accessible algebra \(D/F\) of degree \(n\), where \(F\) has characteristic prime to \(n\) and contains a primitive \(n\)-th root of 1, is cyclic. Together with results of the authors [in Proc. Am. Math. Soc. 84, 162-164 (1981; Zbl 0492.16022)] this leads to a proof that 6-accessible algebras are cyclic. It follows that for \(K\) as before, \(D/K\) is cyclic if \(n\) is a prime \(\leq 7\) and \(D/K\) has a maximal subfield with soluble splitting field.