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I'd like to ask the following question concerning a MAGMA command I'm probably not aware of.

Is it possible to construct a field of the form $F=\mathbb{F}_p(\zeta)$ with MAGMA, where $\zeta$ denotes a primitive $m$-th root of unity in an extension of $\mathbb{F}_p$ with the property that gcd$(p,m)>1$?

I'm particularly interested in the case where $p=2$.

Thanks for the help in advance.

Bernhard Boehmler
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    If $\zeta$ has irreducible degree $k$ over $\mathbb F_p$, then you are asking for the finite field $\mathbb F_{p^k}$. – hardmath Sep 04 '18 at 13:54
  • What did the people who write and maintain the software say? Surely you didn't skip asking the group of people who know the software best? – rschwieb Sep 04 '18 at 13:56
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    And this might be useful? Isn't it just a quotient ring? Surely it isn't hard to do quotients in MAGMA. – rschwieb Sep 04 '18 at 13:57
  • Thank you very mch for the three comments. Thanks for the linked reference. I missed the command SplittingField. Unfortunately, I didn't ask the people developing MAGMA before. So, one could do $Kx:=PolynomialRing(GF(p));$, then $f:=x^m-1;$ and then $L:=SplittingField(f);$. @rschwieb you can post this and / or your linked reference as an answer, if you like. – Bernhard Boehmler Sep 04 '18 at 14:26
  • Please edit: this is ok with [magma-cas] but is not a [magma] question, maybe [fields] if you wanna use 5 tags which is good :) – MattAllegro Sep 13 '18 at 20:21
  • Thank you very much for your comment. I have changed the tags. – Bernhard Boehmler Sep 14 '18 at 21:05

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To give this old question a proper answer (solution is already found in the comments):

function cyclotomic_field(p,m)
    Fp := GF(p);
    Fpx<x> := PolynomialRing(Fp);
    return SplittingField(x^m-1);
end function;

Then for example:

> cyclotomic_field(7,33);
Finite field of size 7^10
azimut
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