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How to Compute $\operatorname{Pic}(\mathbb{Z}[\zeta_p,1/p])$

asked 2020-04-02 10:08:04 -0500

Joel gravatar image

Consider the extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$, where $\zeta_p$ is a primitive $p^{\text{th}}$ root of unity (and $p$ is a prime number). Form the ring of integers $\mathbb{Z}[\zeta_p]$. Now, invert the prime $p$, to obtain the ring $R=\mathbb{Z}[\zeta_p,1/p]$. I want to compute $\operatorname{Pic}(R)$.

Is this possible to do using sagemath? If so, how? How do I construct the ring $R$ using sagemath?

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What's your definition of $\operatorname{Pic}$? You could construct $R$ as a quotient of $\mathbb{Z}[x,y]$.

rburing gravatar imagerburing ( 2020-04-03 03:12:35 -0500 )edit

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answered 2020-04-02 13:16:34 -0500

Sébastien gravatar image

This will be an incomplete answer as I don't know what Pic means. Also, I am not sure if what is below is the proper way to achieve what you want, but at least it may give you some first hints toward the good direction:

sage: K.<xi> = CyclotomicField(13)
sage: L.<t> = K[]
sage: R.<pinv> = L.extension(13*t-1)

This creates those rings:

sage: K
Cyclotomic Field of order 13 and degree 12
sage: L
Univariate Polynomial Ring in t over Cyclotomic Field of order 13 and degree 12
sage: R
Univariate Quotient Polynomial Ring in pinv over Univariate Polynomial Ring in t over Cyclotomic Field of order 13 and degree 12 with modulus 13*pinv - 1

And those elements:

sage: xi
xi
sage: pinv
1/13

which you can work with:

sage: xi^11
xi^11
sage: xi^12
-xi^11 - xi^10 - xi^9 - xi^8 - xi^7 - xi^6 - xi^5 - xi^4 - xi^3 - xi^2 - xi - 1
sage: pinv * xi^10
1/13*xi^10
sage: pinv * xi^10 + 14*pinv*xi^4+pinv^3
1/13*xi^10 + 14/13*xi^4 + 1/2197

Maybe there is a way to get that p instead of my pinv. I let someone else to give a better answer.

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Note the $\mathbb{Z}$ in the question.

rburing gravatar imagerburing ( 2020-04-02 13:28:46 -0500 )edit

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Asked: 2020-04-02 10:08:04 -0500

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Last updated: Apr 02