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If $K/F$ is a Galois extension of fields, a stem field for $K/F$ is a field $E$ such that $F\subseteq E\subseteq K$ and $K$ is the Galois closure of $E/F$.

This is connected to the notion of the stem field of a polynomial. If $f\in F[x]$ is a separable irreducible polynomial of degree $n$ with roots $\alpha_1, \ldots, \alpha_n$ (in some extension field), then the fields $F(\alpha_i)$ are the stem fields of the polynomial $f$. The splitting field of $f$ is $K=F(\alpha_1,\ldots,\alpha_n)$, which is a Galois extension of $F$, and the fields $F(\alpha_i)$ are stem fields for $K/F$ as defined above.

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2019-10-29 16:16:37
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