Euclid's Algorithm in Pure Quartic Fields
Shigeki Egami · Tokyo Journal of Mathematics · 1979
A finite algebraic number field $K$ is said to be euclidean if, for any integers $\alpha$ and $\beta( eq 0)$ of $K$ , there is an integer $\gamma$ of $K$ such that $|N_{K}(\alpha-\beta\gamma)|<|N_{K}\beta|$ .It is well-known that there are exactly 21 quadratic euclidean fields (see E. S. Bernes and H. P. F. Swinnerton-Dyer [1]).As for cubic fields H. Davenport [4] showed that there are only a finite number of euclidean fields which are not totally real.There are several finiteness theorems like this.H. Heilbronn [2], [3], showed that, if $p$ is a prime then the number of cyclic euclidean fields of degree $p$ is finite.H. Davenport [5] (cf.J. W. S. Cassels [6]) also proved the finiteness of the number of totally imaginary quartic euclidean fields.In this paper we shall prove the following THEOREM.There exist only a finite number of quartic euclidean fields of the form $Q(\sqrt[4]{m})$ , where $m$ is $a$ 4th power-free rational integer not expressible as 2 $p^{2}$ with a prime $p\equiv 3(mod 8)$ .In proving Theorem we can restrict our consideration to some special forms of quartic fields.Indeed for the fields $Q(\sqrt[4]{-m})$ , where $m$ is a positive integer, the finiteness follows from the result of Davenport mentioned above.Further C. J. Parry [7] proved that the class number of the field $Q(\sqrt[4]{m})$ with a positive integer $m$ is even except those of the following forms (I) $Q(\sqrt[4]{p})p\equiv 5(mod 8),$ $Q(\sqrt[4]{4p})p\equiv 5(mod 8)$ , (II) $Q(\#\overline{p})p\equiv 3(mod 8),$ $Q(\sqrt[4]{2p})p\equiv 3(mod 8)$ , $Q(\sqrt[4]{4p})p\equiv 3,7(mod 8),$ $Q(\sqrt{8p})p\equiv 3(mod 8)$ , (III) $Q(\sqrt[4]{2p^{2}})p\equiv 3(mod 8),$ $Q(\forall\overline{2})$ , where $p$ is a rational prime.Thus our theorem is reduced to the state- ment that the number of euclidean fields of the form (I) or (II) is finite, since an algebraic number field of class number greater than one is not