Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees m and n is again irreducible if and only if m and n are coprime and the involved diamond product satisfies a special cancellation property, the so-called conjugate cancellation. This completes the characterization of irreducible composed products, considered in several previous papers. More generally, we give precise criteria when a diamond product satisfies conjugate cancellation. For diamond products defined via bivariate polynomials, we prove simple criteria that characterize when conjugate cancellation holds. We also provide efficient algorithms to check these criteria. We achieve stronger results as well as more efficient algorithms in the case that the polynomials are bilinear. Lastly, we consider possible constructions of normal elements using composed products and the methods we developed.

Brawley 和 Carlitz 于 1987 年引入了有限域元素的金刚石积和相关的多项式组合积。组合积产生一种从较低次的不可约多项式构造大复合度的不可约多项式的方法。我们表明，当且仅当 m 和 n 是互质并且所涉及的金刚石乘积满足特殊的抵消特性，即所谓的共轭抵消时，两个度数 m 和 n 的不可约多项式的组合乘积再次是不可约的。这完成了之前几篇论文中讨论的不可约组合产物的表征。更一般地说，当钻石产品满足共轭抵消时，我们会给出精确的标准。对于通过二元多项式定义的金刚石产品，我们证明了表征共轭抵消何时成立的简单标准。我们还提供高效的算法来检查这些标准。在多项式为双线性的情况下，我们获得了更强的结果和更高效的算法。最后，我们考虑了使用组合产品和我们开发的方法构建法线元素的可能构造。