Let $K= \mathbb{Q}_3$ and $L = \mathbb{Q}_3(\alpha)$ be defined by $\min_K(\alpha) = x^4 - 3x^2 +18$. Furthermore, let $F$ be the unramified extension of $K$ of degree $4$ (which is generated by a primitive $5$-th root of unity). Now I would like to implement the composite field $FL$ of $F$ and $L$.
I tried it with the following MAGMA code:
K := pAdicField(3,30);
R<x> := PolynomialRing(K);
f := x^4-3*x^2+18;
L := LocalField(K,f);
g := R ! CyclotomicPolynomial(5);
F := LocalField(K,g);
FL := Composite(F,L);
Unfortunately, this only gives me the following error:
>> FL := Composite(F,L);
^
Runtime error in 'Composite': Bad argument types
Argument types given: RngLocA, RngLocA
I don't understand what I did wrong since in the Handbook of MAGMA functions (Volume 4, page 1274) it says
and both $L$ and $F$ are both normal (even Galois) extensions of $K$.
Could you tell me what went wrong and what I could do instead to get the composite field?
