Mathematics Asked by HyperPro on December 15, 2020

I have to proof that the only (field) automorphism of $mathbb{Q}(sqrt d)$ fixing $mathbb{Q}$ are $id$ and the conjugation $sigma$.

I know for every such automorphism $tau$ we have that $$tau(0)=0 \ tau(1)=1 \ tau(-a)=-tau(a) \ tau(a^{-1})=tau(a)^{-1}$$

and it is easy to see that satisfy all these properties. Any other automorphism I think about always violates any of these rules. But this does not mean that there could any very special and weird automorphism I just can not think about

An automorphism must fix $mathbb Q$, since it fixes $1$, and thus all integers, and thus all inverses of integers, and thus all products of integers and inverses of integers, which covers all rational numbers.

Now consider the polynomial $f=X^2-d$. If $f(alpha)=0$, then $f(tau(alpha))=tau(f(alpha))=tau(0)=0$, so any automorphism has to send roots of $f$ to roots of $f$. That is, $sqrt d$ is sent to itself (resulting in the identity automorphism), or to $-sqrt d$ (resulting in the conjugation).

Correct answer by Vercassivelaunos on December 15, 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

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

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