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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

Anonymous

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

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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

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

nbruin gravatar image

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

Try:

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 $\mathbb{Q}_7$.

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

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Last updated: Mar 08