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Let $K/F$ is finite separable extension of fields with Galois closure $L$ and let let $G= \Gal(L/F)$. Then any subfield of $L$ is a resolvent field for $K/F$. These are in bijection with the subgroups of $G$.

One way to group these fields is to collect together subfields with the same Galois closure since given one such field, the other fields are in a collection are visible as siblings of the field. A natural representative of such a collection would be a field of minimal degree over $F$ with ties broken by taking a field whose Galois closure has Galois group $nTj$ with minimal $j$.

In group theoretic terms, $G\leq S_n$ where $n=[K:F]$. The resolvents of $G$ correspond to quotients $G/N$ where $N \unlhd G$. For each normal subgroup $N$, we then look for $H\leq G$ such that the core of $H$ equals $N$, and then pick the subgroup $H$ of minimal index in $G$ so that when $G$ acts transitively on the $m$ left cosets of $H$ by left translation, the T-number of the permutation reputation is minimized.

The fixed field of $H$ is then a stem field for $L^N$, the fixed field of $N$ over $F$. The Galois group for $L^N/F$ has is a subgroup of $S_m$, up to conjugation.

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2019-09-15 18:46:57
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