Kronecker classes of algebraic number fields

Kronecker classes of field extensions of small degree

“…(a) n = 5, G(3f)=A 5 , U^A A , V~S 3 The Kronecker equivalence of two extensions L and K of k is equivalent to a group theoretic condition; namely if M : k is a Galois extension containing L and K as intermediate fields and if G is the Galois group of M : k and U, V are the fixed groups of K and L respectively, then it is shown in [3, §1] that K and L are Kronecker equivalent over k if and only if U G = V G where, for a subgroup H of G, H G denotes the set theoretic union \J g€G H g . This group theoretic condition is studied in Section 2; the group theoretic analogue, Theorem 2, of Theorem 1, is proved there and Theorem 1 is deduced from it.…”

“…Hence V contains an element h of type 2 2 moving the point 5. Since V contains no elements of type 2 3 and since h centralizes g we must have h = (12)(56) so that V contains (56). Similarly V contains (34) and hence V contains (12)(34)(56) which is a contradiction.…”

“…Suppose first that UnB = {1}. Then U < 5 3 . If U had order 2 or 3 then U and V would be equal so we may assume that U ~ 5 3 and hence that A is Z 2 x S 4 or GL (2,3).…”

“…Then [8] there are at least two conjugacy classes of permutations of type 2 in B and hence if V fixes a point then V = U. So assume that V f)B = V 3 …”

Kronecker conjugacy of polynomials

“…See [19], [17], [12, Section 19.5]. This condition has also been investigated (independently of its number theoretical context) by various group theorists.…”