Mathematics Asked by Confusion on December 7, 2020

Here is the question I want to answer:

Show that $mathbb{Z}[sqrt{3}]$ is dense in $mathbb{R}.$

I got this hint:

Hint: Find $u in mathbb{Z}[sqrt{3}]^*$ with $u > 1$ and consider $lim_{nrightarrow infty} u^{-n}.$

**My question is:**

How can we find this $zeta$? how can I prove that it is a unit? and why we are taking it from the set of units?should this unit be greater than 1 or less than 1?

You don't really need a unit $zeta$, just an element $zeta$ such that $0 < zeta < 1$. You can start with any element $r$ in your ring that is not integral. Then $zeta ={r} = r - [r]$ is in the interval $(0,1)$, and can be used in your nice argument.

$bf{Added:}$ For example, consider $r=sqrt{3}$. We have $[r]=1$. Take $zeta= sqrt{3}-1$, in $(0,1)$, but not a unit ( since its norm $(sqrt{3}-1)(sqrt{3}+1)=2>1$).

Correct answer by orangeskid on December 7, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP