TY - JOUR
T1 - The lattice of subfields of a radical extension
AU - de Orozco, MariáAcosta
AU - Vélez, William Yslas
PY - 1982/12
Y1 - 1982/12
N2 - Let xm - a be irreducible over F with char F{does not divide}m and let α be a root of xm - a. The purpose of this paper is to study the lattice of subfields of F(α) F and to this end C( F(α) F, k) is defined to be the number of subfields of F(α) of degree k over F. C( F(α) F, pn) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then C( F(α) F, k) = C( F(α) F, (k, n)) = C( N F, (k, n)). The irreducible binomials xs - b, xs - c are said to be equivalent if there exist roots βs = b, γs = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x2e - a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.
AB - Let xm - a be irreducible over F with char F{does not divide}m and let α be a root of xm - a. The purpose of this paper is to study the lattice of subfields of F(α) F and to this end C( F(α) F, k) is defined to be the number of subfields of F(α) of degree k over F. C( F(α) F, pn) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then C( F(α) F, k) = C( F(α) F, (k, n)) = C( N F, (k, n)). The irreducible binomials xs - b, xs - c are said to be equivalent if there exist roots βs = b, γs = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x2e - a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.
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U2 - 10.1016/0022-314X(82)90040-3
DO - 10.1016/0022-314X(82)90040-3
M3 - Article
AN - SCOPUS:49049139651
SN - 0022-314X
VL - 15
SP - 388
EP - 405
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 3
ER -