conjugates of algebraic numbers

zverik136 · #1 $Old$ October 26th, 2009, 12:56 AM

hello,
i am having a problem with the following question:

Let $\alpha \in \mathbb{C}$ be an algebraic number of degree $d$ and denote by $\alpha^{(1)}, \ldots, \alpha^{(d)}$ the conjugates of $\alpha$.

a. Let $m = den(\alpha)$, ie the smallest positive integer $m \in \mathbb{Z}$ such that $m\alpha$ is an algebraic integer. Prove that $m^d \alpha^{(1)} \ldots \alpha^{(d)} \in \mathbb{Z}$.

b. Suppose that $\alpha \neq 0$. Prove that $|\alpha| \geq den(\alpha)^{-d} |\bar{\alpha}|^{1-d}$, where $\bar{\alpha}$ is the house of $\alpha$.

c. Using b. give a proof of the following inequality of Liouville (1844): Let $\alpha$ be an algebraic number in $\mathbb{R}$ of degree $d /geq 2$. Then there is a constant $c(\alpha) > 0$ such that $ |\alpha - p/q| \geq c(\alpha) q^{-d}$ for all $p, q \in \mathbb{Z}$ with $q>0$.

I thought I would write out the whole thing for clarity

Thank you

tonio · #2 $Old$ October 26th, 2009, 03:56 AM

Quote:

Originally Posted by zverik136 $View Post$

hello,
i am having a problem with the following question:

Let $\alpha \in \mathbb{C}$ be an algebraic number of degree d and denote by $\alpha^{(1)}, \ldots, \alpha^{(d)}$ the conjugates of $\alpha$ .

a. Let $m = den(\alpha)$ , ie the smallest positive integer $m \in \mathbb{Z}$ such that $m\alpha$ is an algebraic integer. Prove that $m^d \alpha^{(1)} \ldots \alpha^{(d)} \in \mathbb{Z}$ .

b. Suppose that $\alpha \neq 0$ . Prove that $|\alpha| \geq den(\alpha)^{-d} |\bar{\alpha}|^{1-d}$ , where $\bar{\alpha}$ is the house of $\alpha$ .

c. Using b. give a proof of the following inequality of Liouville (1844): Let $\alpha$ be an algebraic number in $\mathbb{R}$ of degree $d /geq 2$ . Then there is a constant $c(\alpha) > 0$ such that $|\alpha - p/q| \geq c(\alpha) q^{-d}$ for all $p, q \in \mathbb{Z}$ with $q>0$ .

I thought I would write out the whole thing for clarity

Thank you

Hi there. I tried to correct your LaTex text as much as I could since in this site, unfortunately, to write properly in LaTex we have to open sentences with [math] and close them with [/math], instead of the well known and simpler $.
Nevertheless, as you can see, somethings still didn't come out nicely, as that thing "den( $\alpha$ )" and "d /geg2"

Tonio

Bruno J. · #3 $Old$ October 26th, 2009, 07:19 PM

Quote:

Originally Posted by zverik136

hello,
i am having a problem with the following question:

Let be an algebraic number of degree d and denote by the conjugates of .

a. Let , ie the smallest positive integer such that is an algebraic integer. Prove that .

b. Suppose that . Prove that , where is the house of .

c. Using b. give a proof of the following inequality of Liouville (1844): Let be an algebraic number in of degree . Then there is a constant such that for all with .

I thought I would write out the whole thing for clarity

Thank you

Hello zverik!

Let $\sigma_1,\hdots, \sigma_d$ be the embeddings of $\mathbb{Q}(\alpha)$ in $\mathbb{C}$ .

Since $m\alpha$ is an algebraic integer, so are its conjugates $m\alpha^{(1)},\hdots,m\alpha^{(d)}$ , and hence so is the product of its conjugates $P=m^d\alpha^{(1)}\hdots\alpha^{(d)}$ . But clearly $P$ is fixed by all of $\sigma_1,\hdots,\sigma_d$ , so it is in $\mathbb{Q}$ ; since the only rational algebraic integers are the usual integers $\mathbb{Z}$ , we have $P\in \mathbb{Z}$ .

I do not understand b), so I can't really do c) unless I try another path. What is the house of an algebraic number?

zverik136 · #4 $Old$ October 27th, 2009, 12:52 AM

hello

thank you for your reply!

the house of $\alpha$ is defined to be $\bar{\alpha} = \max (|\alpha^{(1)}|, \ldots , |\alpha^{(d)}|)$ .

tonio · #5 $Old$ October 27th, 2009, 05:00 AM

Quote:

Originally Posted by zverik136 $View Post$

hello

thank you for your reply!

the house of $\alpha$ is defined to be $\bar{\alpha} = \max (|\alpha^{(1)}|, \ldots , |\alpha^{(d)}|)$ .

I'm afraid we still don't know what is $den(\alpha)$

Tonio

zverik136 · #6 $Old$ October 27th, 2009, 05:51 AM

The only definition of den i was given was as above:

The denominator of $\alpha$ , denoted by $den(\alpha)$ , is the smallest positive integer $m \in \mathbb{Z}$ such that $m\alpha$ is an algebraic integer.

thank you