Applications of arithmetic theory of algebraic groups to computational number theory

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In this research we investigate a generalization of Kummer theory to fields without roots of unity. Kummer theory is a basic tool in algebra and number theory and has many applications in these areas. Our generalization uses commutative algebraic groups called norm tori. Under certain natural conditions, we prove a Kummer duality induced from a self-isogeny of norm tori. This is a natural extension of the classical Kummer theory. It also describes the cyclic extensions over certain fields without roots of unity.As an application, we develop a method to compute cyclic polynomials arising from our Kummer theory and calculate some example of such polynomials using computer algebra system MAGMA. In the case of quintic cyclic polynomial, we can show a relationship between our Kummer polynomials and classical Lehmer polynomials. This enables us to give a complete description of the decomposition law in the cyclic quintic extensions.As our result is quite general in the nature, we can expect more applications in the area of algebra and number theory.