This is something I've been trying to work on on my own for the past few days but I'm not sure how to approach it.(adsbygoogle = window.adsbygoogle || []).push({});

My Question:

a. Let E be a Galois extension of a field F with characteristic 0. Prove that there is a unique smallest subfield K such that [itex] F \subseteq K \subseteq E [/itex], K is normal over F and E is subradical over K.

[One needs the following result: Let [itex]A[/itex] and [itex]B[/itex] be solvable subgroups of a group [itex]G[/itex] and suppose that [itex]A[/itex] is normal in [itex]G[/itex]. Then [itex]AB[/itex] is solvable.]

b. Let f be an irreducible polynomial over [itex]\mathbb{Q}[/itex] which has degree 5 and at least two complex roots. Prove that [itex]Gal(f)[/itex] has order 10, 20, 60, or 120.

Thanks in advance for any kind of direction that you can provide me with...

**Physics Forums | Science Articles, Homework Help, Discussion**

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!

# Galois extension of a field with Characteristic 0

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**