### Abstract

If K is a field and char K\n, then any binomial x^{n} - b ϵ K[x] has the property that K(a) is its splitting field for any root a iff a primitive nth root of unity an is an element of K. Thus, if aneK, any irreducible binomial x^{n} - b ϵ K[x] is automatically normal. Similar nice results about binomials x^{n} – b (Kummer theory comes to mind) can be obtained with the assumption l ϵ K. In this paper, without assuming the appropriate roots of unity are in K, one asks: what are the binomials x^{m} - b ϵ K[x] having the property that K(a) is its splitting field for some root al Such binomials are called partially normal. General theorems are obtained in case K is a real field. A complete list of partially normal binomials together with their Galois groups is found in case K–Q, the rational numbers.

Original language | English (US) |
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Pages (from-to) | 403-417 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 72 |

Issue number | 2 |

Publication status | Published - 1977 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*72*(2), 403-417.