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Can't construct automorphisms of p-adic fields

asked 2015-09-02 17:40:36 -0500

siggytm gravatar image

I'm trying to construct automorphisms of finite extensions of $\mathbb Q_p$ and getting a funny error. Here's a prototypical example:

Evaluating the cyclotomic polynomial $x^4 + x^3 + x^2 + x + 1$ at $x+1$ gives an Eisenstein polynomial for the prime $p=5$. If $\pi$ is a root of $(x+1)^4 + (x+1)^3 + (x+1)^2 + (x+1) + 1$ then $\pi+1$ will be a primitive 5th root of unity.

K.<pi> = Qp(5).ext(sum((1+x)^i for i in range(5)))

So $K=\mathbb Q_p(\zeta_5)$ with uniformizer $\pi=\zeta_5-1$. Now I want to define the automorphism $\tau:\zeta_5\mapsto\zeta_5^2$. We have $\tau(\pi)=(1+\pi)^2-1=2\pi+\pi^2$. But the following

tau = K.hom([2*pi+pi^2])

results in the error TypeError: images do not define a valid homomorphism. What's going on? Are hom's of $p$-adic fields not really implemented yet or am I doing something wrong?

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answered 2015-09-03 11:11:38 -0500

nbruin gravatar image

You're correct. In your example you end up executing:

sage: sage.rings.morphism.RingHomomorphism_im_gens(K.Hom(K), [2*pi+pi^2], check=true)
NotImplementedError: Verification of correctness of homomorphisms from Eisenstein Extension of 5-adic Field with capped relative precision 20 in pi defined by (1 + O(5^20))*x^4 + (5 + O(5^20))*x^3 + (2*5 + O(5^20))*x^2 + (2*5 + O(5^20))*x + (5 + O(5^20)) not yet implemented.

but then this error gets caught in wrapping code and recast as a less informative error. (rant: this is a general problem in Python: people think they make code robust by putting try/except statements in but in reality they just reduce the ability of the code to properly report the problem)

Once you know the problem, the workaround is easy:

sage: tau = K.hom([2*pi+pi^2],check=false)
sage: tau
Ring endomorphism of Eisenstein Extension of 5-adic Field with capped relative precision 20 in pi defined by (1 + O(5^20))*x^4 + (5 + O(5^20))*x^3 + (2*5 + O(5^20))*x^2 + (2*5 + O(5^20))*x + (5 + O(5^20))
  Defn: pi + O(pi^77) |--> 2*pi + pi^2 + O(pi^77)
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Thank you! That's a nice easy workaround.

siggytm gravatar imagesiggytm ( 2015-09-03 12:06:19 -0500 )edit

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Asked: 2015-09-02 17:40:36 -0500

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Last updated: Sep 03 '15