(adsbygoogle = window.adsbygoogle || []).push({}); Question:I need to show that [itex]K = \mathbb{Q}(i, 2^{1/4}) [/itex] is a Galois extensions of [itex]\mathbb{Q}[/itex].

If I show that [itex]|Gal(\mathbb{Q}(i, 2^{1/4})/\mathbb{Q})|= [\mathbb{Q}(i, 2^{1/4}):\mathbb{Q}] [/itex], then we're done. Another approach is to find an irreducible polynomial [itex]f(x)\in \mathbb{Q}[x][/itex] such that K is the splitting field for [itex]f[/itex], then we're done.

I first considered [itex] f(x) = x^4 -2 [/itex] but this is a degree 4 polynomial. I'm looking for an irreducible degree 8 polynomial. How do I find such polynomial given K? What's the most efficient way to find such polynomial?

Note that K is a Galois extension over the rational numbers if K is the splitting field for some polynomial f and f does not split completely into linear factors over any proper subfield of K containing [itex]\mathbb{Q}[/itex].

Thank you!

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# An elementary question regarding Galois theory

Loading...

Similar Threads for elementary question regarding | Date |
---|---|

B Proof of elementary row matrix operation. | Jun 6, 2017 |

I Intuition behind elementary operations on matrices | May 20, 2017 |

B Elementary eigenvector question | Mar 27, 2017 |

Elementary question about Dirac notation | Mar 22, 2013 |

One more elementary question, on square roots | Dec 22, 2006 |

**Physics Forums - The Fusion of Science and Community**