Ask Your Question

How to define $\mathbb{Q}_p(\sqrt{5})$ and $\mathbb{Q}_p (\sqrt{5} \sqrt{3})$ and find their valuation rings for $p=7$?

asked 2019-03-06 22:03:29 -0500

anonymous user


updated 2019-03-07 02:48:20 -0500

slelievre gravatar image

I tried finite extension $\mathbb({Q}_p$ I unable do it. It will also great if help me with how to define the valuation ring of that finite extension

edit retag flag offensive close merge delete


Did you mean $\mathbb{Q}_p (\sqrt{5}, \sqrt{3})$ instead of $\mathbb{Q}_p (\sqrt{5} \sqrt{3})$?

slelievre gravatar imageslelievre ( 2019-03-10 13:48:50 -0500 )edit

1 answer

Sort by ยป oldest newest most voted

answered 2019-03-08 12:06:39 -0500

nbruin gravatar image

updated 2019-03-08 15:52:42 -0500


sage: Qp=pAdicField(7)
sage: R.<x>=Qp[]
sage: K.<a>=Qp.extension(x^2-5)
sage: OK=K.integer_ring()
sage: OK
7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$, so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$.

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower


Asked: 2019-03-06 22:03:29 -0500

Seen: 54 times

Last updated: Mar 08