ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 05 Mar 2017 17:44:55 +0100ramified extension of Qphttps://ask.sagemath.org/question/36595/ramified-extension-of-qp/ I am wondering if anyone knows
1. How to define Q_p[\mu_p] in sage, where \mu_p is a primitive p-th root of unity in \bar Q_p.
2. How to write b=\mu_p.
3. Solve a quadratic equation x^2+a_1x+a_2=0, where a_1 and a_2 are both in Q_p[\mu_p].
4. Is log function defined on Q_p[\mu_p] .
Thank you in advance!Mon, 13 Feb 2017 04:52:01 +0100https://ask.sagemath.org/question/36595/ramified-extension-of-qp/Answer by dan_fulea for <p>I am wondering if anyone knows
1. How to define Q_p[\mu_p] in sage, where \mu_p is a primitive p-th root of unity in \bar Q_p.
2. How to write b=\mu_p.
3. Solve a quadratic equation x^2+a_1x+a_2=0, where a_1 and a_2 are both in Q_p[\mu_p].
4. Is log function defined on Q_p[\mu_p] .</p>
<p>Thank you in advance!</p>
https://ask.sagemath.org/question/36595/ramified-extension-of-qp/?answer=36836#post-id-36836I tried:
sage: p = 11
sage: K = Qp( p )
sage: R.<x> = PolynomialRing( K )
sage: L.<b> = K.extension( x^p - p )
sage: b
b + O(b^221)
sage: b.norm()
11 + O(11^21)
sage: -log( 1-b ) + O(b^12)
1 + b + 6*b^2 + 4*b^3 + 3*b^4 + 9*b^5 + 2*b^6 + 8*b^7 + 7*b^8 + 5*b^9 + 10*b^10 + 6*b^11 + O(b^12)
# But...
# sage: var( 'y' );
# sage: solve( y^2 - b*y + 7 == 0 , y ) # --> error
Sun, 05 Mar 2017 17:44:55 +0100https://ask.sagemath.org/question/36595/ramified-extension-of-qp/?answer=36836#post-id-36836