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ramified extension of Qp

asked 2017-02-12 21:52:01 -0500

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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!

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answered 2017-03-05 10:44:55 -0500

dan_fulea gravatar image

I 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
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Asked: 2017-02-12 21:52:01 -0500

Seen: 17 times

Last updated: Mar 05