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In this paper, we consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. We present the characterization of some known field extensions in terms of polynomial composites. This paper contains the open problem of characterization of ideals in polynomial composites with respect to various field extensions. We also present the full possible characterization of certain field extensions. Moreover, in this paper we show that any finite group is a Galois group of some field extensions and present the inverse Galois problem solved.

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≅ ℤ2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups D* is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and is divisible. [ABSTRACT FROM AUTHOR]

A fast algorithm for the computation of a cosine‐like number‐theoretic transform is presented. The method, which corresponds to a finite field extension of a method originally designed for computing real‐valued discrete cosine transforms, is recursive and suitable for VLSI implementation. A general flow diagram for the proposed algorithm is given and shows that, in some specific cases, it can be evaluated using additions and bit‐shift operations only. [ABSTRACT FROM AUTHOR]