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Abstract. We study polynomials giving cyclic extensions over rational function fields with one variable satisfying some conditions. By using them, we construct families of cyclic polynomials over some algebraic number fields. And these families give non-Kummer (or non-Artin-Schreier) cyclic extensions. In this paper, we see that our polynomials have two nice arithmetic properties. One is simplicity: our polynomials and their discriminants have more simple expressions than previous results, e.g. Dentzer (1995), Malle and Mazat (1999) and Smith (1991), etc. The other is a "systematic" property: if one of our polynomials f gives an extension L/K, then for every intermediate field M we can easily find polynomials giving M/K from f systematically.

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We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.

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