Abstract. We give the complete stably rational classification of algebraic tori of dimensions 4 and 5 over a field k. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank 4 and 5 is given. We show that there exist exactly 487 (resp. 7, resp. 216) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 4, and there exist exactly 3051 (resp. 25, resp. 3003) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 5. We make a procedure to compute a flabby resolution of a G-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a G-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby G-lattices of rank up to 6 and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for G-lattices holds when the rank ≤ 4, and fails when the rank is 5. Indeed, there exist exactly 11 (resp. 131) G-lattices of rank 5 (resp. 6) which are decomposable into two different ranks. Moreover, when the rank is 6, there exist exactly 18 G-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that H 1 (G, F ) = 0 for any Bravais group G of dimension n ≤ 6 where F is the flabby class of the corresponding G-lattice of rank n. In particular, H 1 (G, F ) = 0 for any maximal finite subgroup G ≤ GL(n, ) where n ≤ 6. As an application of the methods developed, some examples of not retract (stably) rational fields over k are given.
Abstract. Let k be any field, G be a finite group acting on the rational function field k(xg : g ∈ G) by h · xg = x hg for any h, g ∈ G. Define k(G) = k(xg : g ∈ G) G . Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if (G) is rational over , then B 0 (G) = 0 where B 0 (G) is the unramified Brauer group of (G) over . Bogomolov showed that, if G is a p-group of order p 5 , then B 0 (G) = 0. This result was disproved by Moravec for p = 3, 5, 7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p 5 . Then B 0 (G) = 0 if and only if G belongs to the isoclinism family Φ 10 in R. James's classification of groups of order p 5 .
Let k be a field of characteristic = 2. We give an answer to the field intersection problem of quartic generic polynomials over k via formal Tschirnhausen transformation and multi-resolvent polynomials.
Let K be a field of characteristic not two and K(x, y, z) the rational function field over K with three variables x, y, z. Let G be a finite group acting on K(x, y, z) by monomial K-automorphisms. We consider the rationality problem of the fixed field K(x, y, z) G under the action of G, namely whether K(x, y, z) G is rational (that is, purely transcendental) over K or not. We may assume that G is a subgroup of GL(3, Z) and the problem is determined up to conjugacy in GL(3, Z). There are 73 conjugacy classes of G in GL(3, Z). By results of Endo-Miyata, Voskresenskiȋ, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in GL(3, Z) have negative answers to the problem under certain monomial actions over some base field K, and the necessary and sufficient condition for the rationality of K(x, y, z) G over K is given. In this paper, we show that the fixed field K(x, y, z) G under monomial action of G is rational over K except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over K. For the unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if K is quadratically closed field then K(x, y, z) G is rational over K. We also give an application of the result to 4-dimensional linear Noether's problem.
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