# Galois Extension

• Apr 12th 2010, 02:52 PM
mathman88
Galois Extension
Let $\displaystyle F$ be a subfield of $\displaystyle \mathbb{R}$. Let $\displaystyle a$ be an element of $\displaystyle F$ and let $\displaystyle K=F(\sqrt[n]{a})$ where $\displaystyle \sqrt[n]{a}$ denotes the real $\displaystyle n^{th}$ root of $\displaystyle a$.

Prove thatif $\displaystyle L$ is any Galois extension of $\displaystyle F$ contained in $\displaystyle K$ then $\displaystyle [L:F]\leq2$.

I have no idea how to approach this problem.

thanks everyone!
• Apr 12th 2010, 08:32 PM
FancyMouse
First, you need to know that the minimal polynomial of $\displaystyle \sqrt[n]{a}$ over F is $\displaystyle x^d-b$ for some $\displaystyle d|n,b\in F$ (not a hard lemma)
Then if L is Galois over F, we must have $\displaystyle \zeta_d\in F$ in order to throw all conjugates of $\displaystyle \sqrt[d]{b}$ into K. But clearly $\displaystyle \zeta_d$ is not real unless $\displaystyle d\leq2$.
• Apr 13th 2010, 02:34 AM
tonio
Quote:

Originally Posted by FancyMouse
First, you need to know that the minimal polynomial of $\displaystyle \sqrt[n]{a}$ over F is $\displaystyle x^d-b$ for some $\displaystyle d|n,b\in F$ (not a hard lemma)
Then if L is Galois over F, we must have $\displaystyle \zeta_d\in F$ in order to throw all conjugates of $\displaystyle \sqrt[d]{b}$ into K. But clearly $\displaystyle \zeta_d$ is not real unless $\displaystyle d\leq2$.

Remark: By $\displaystyle \zeta_d$ Fancymouse most probably meant a primitive d-th root of unity in $\displaystyle \mathbb{C}$

Tonio
• Apr 13th 2010, 11:21 AM
mathman88
Quote:

Originally Posted by FancyMouse
First, you need to know that the minimal polynomial of $\displaystyle \sqrt[n]{a}$ over F is $\displaystyle x^d-b$ for some $\displaystyle d|n,b\in F$ (not a hard lemma)
Then if L is Galois over F, we must have $\displaystyle \zeta_d\in F$ in order to throw all conjugates of $\displaystyle \sqrt[d]{b}$ into K. But clearly $\displaystyle \zeta_d$ is not real unless $\displaystyle d\leq2$.

How is that lemma proven?

Thanks!
• Apr 13th 2010, 02:36 PM
FancyMouse
Quote:

Originally Posted by mathman88
How is that lemma proven?

Thanks!

Sorry. I mixed up with something else. The lemma'' I stated before is not only unnecessary but also wrong. The correct version of the lemma needs the assumption that $\displaystyle \zeta_n\in F$. The proof of the correct version needs Hilbert Theorem 90 using norms (or traces if n divides the characteristic of F), which is not hard but nontrivial to prove.

However, we don't even need this. $\displaystyle \alpha$ is a root of x^n-a implies that all roots are of the form $\displaystyle \zeta_n^k\sqrt[n]{a}$ for some k (but not the converse, i.e. not all things of this form are the roots of x^n-a), since the minimal polynomial of $\displaystyle \sqrt[n]{a}$ must divide x^n-a. But use my idea before, that only two of the $\displaystyle \zeta_n^k$ can be real. This implies that $\displaystyle \sqrt[n]{a}$ can only have at most two conjugates, hence the degree is <= 2
• Apr 13th 2010, 03:04 PM
mathman88
Quote:

Originally Posted by FancyMouse
But use my idea before, that only two of the $\displaystyle \zeta_n^k$ can be real. This implies that $\displaystyle \sqrt[n]{a}$ can only have at most two conjugates, hence the degree is <= 2

Sorry, but I don't see why this is always true. I see why only up to 2 $\displaystyle \zeta_n^k$ can be real, but I don't see why this implies that $\displaystyle \sqrt[n]{a}$ can only have at most two conjugates, hence the degree is <= 2.