### Abstract

Let x^{m} - a be irreducible over F with char F{does not divide}m and let α be a root of x^{m} - 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, p^{n}) 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 x^{s} - b, x^{s} - c are said to be equivalent if there exist roots β^{s} = b, γ^{s} = 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 x^{2e} - a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.

Original language | English (US) |
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Pages (from-to) | 388-405 |

Number of pages | 18 |

Journal | Journal of Number Theory |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1982 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*15*(3), 388-405. https://doi.org/10.1016/0022-314X(82)90040-3