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I would like to construct a field (NumberField) that contains a (primitive) $n$-th root of unity and $i = \sqrt{-1}$ using a computer algebra system MAGMA, i.e. $\mathbb{Q}(\zeta_n, i)$.

I tried constructing it as follows (for $n=7$):

n := 7;
Qx<x> := PolynomialRing(Rationals());
f := CyclotomicPolynomial(n);
g := x^2+1;
F<a> := SplittingField([f,g])
DefiningPolynomial(F);

Here the element a gives me the $n$-th root of unity. But how do I get b such that $b^2=-1$?

Thanks in advance! -- Mike

P.S. If someone wants to test MAGMA, you can try online here.

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  • $\begingroup$ You can create E<a> := SplittingField(f); and then F<i> := SplittingField(g,F); $\endgroup$ – Morgan Rodgers Mar 30 at 16:47
  • $\begingroup$ You can also create F as a cyclotomic field, F<a> := CyclotomicField(LCM(n,4)); and then ask for RootOfUnity(4,F); $\endgroup$ – Morgan Rodgers Mar 30 at 16:50
  • $\begingroup$ @MorganRodgers Thanks. I couldnt think of LCM command. Also I didnt know how to compute SplittingField over another field. $\endgroup$ – Mike V.D.C. Apr 1 at 12:50

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