# Galois subfields of real radical extensions are at most quadratic

###### Theorem 1.

Suppose $F\mathrm{\subset}L\mathrm{\subset}K\mathrm{=}F\mathit{}\mathrm{(}\sqrt[n]{\alpha}\mathrm{)}\mathrm{\subset}\mathrm{R}$ are fields with $\alpha \mathrm{\in}F$ and $L$ Galois over $F$. Then $\mathrm{[}L\mathrm{:}F\mathrm{]}\mathrm{\le}\mathrm{2}$.

Proof.
Let ${\zeta}_{n}$ be a primitive ${n}^{\mathrm{th}}$ root of unity^{}, and define ${F}^{\prime}=F({\zeta}_{n})$, ${L}^{\prime}=L({\zeta}_{n})$, and ${K}^{\prime}=K({\zeta}_{n})={F}^{\prime}(\sqrt[n]{\alpha})$.

$$\text{xymatrix}\mathrm{@}R1pc\mathrm{@}C.3pc\mathrm{\&}{K}^{\prime}=K({\zeta}_{n})={F}^{\prime}(\sqrt[n]{\alpha})\text{ar}\mathrm{@}-[dl]\text{ar}\mathrm{@}-[dr]\mathrm{\&}\mathrm{\&}K=F(\sqrt[n]{\alpha})\text{ar}\mathrm{@}-[dr]\mathrm{\&}\mathrm{\&}{L}^{\prime}=L({\zeta}_{n})\text{ar}\mathrm{@}-[dl]\text{ar}\mathrm{@}-[dr]\mathrm{\&}\mathrm{\&}L\text{ar}\mathrm{@}-[dr]\mathrm{\&}\mathrm{\&}{F}^{\prime}=F({\zeta}_{n})\text{ar}\mathrm{@}-[dl]\mathrm{\&}\mathrm{\&}F\mathrm{\&}$$ |

Now, ${L}^{\prime}/{F}^{\prime}$ is Galois since $L/F$ is. But ${K}^{\prime}$ is a Kummer extension^{} of ${F}^{\prime}$, so has cyclic Galois group^{} and thus ${L}^{\prime}/{F}^{\prime}$ has cyclic Galois group as well (being a quotient^{} of $\mathrm{Gal}({K}^{\prime}/{F}^{\prime})$). Thus ${L}^{\prime}$ is a Kummer extension of ${F}^{\prime}$, so that ${L}^{\prime}={F}^{\prime}(\sqrt[n]{\beta})$ for some $\beta \in {F}^{\prime}$. It follows that $L=F(\sqrt[n]{\beta})$. But since $L$ is Galois over $F$, it follows that $n\le 2$ (since otherwise in order to be Galois, $L$ would have to contain the non-real ${n}^{\mathrm{th}}$ roots of unity).

Title | Galois subfields of real radical extensions are at most quadratic |
---|---|

Canonical name | GaloisSubfieldsOfRealRadicalExtensionsAreAtMostQuadratic |

Date of creation | 2013-03-22 17:43:05 |

Last modified on | 2013-03-22 17:43:05 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 7 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 12F10 |

Classification | msc 12F05 |