Restriction map of Galois groups

galois-theory

I am trying to understand the meaning of the restriction map of Galois groups. Suppose $F\subset L_1\subset L_2\subset K$ are fields such that $L_1$ and $L_2$ are Galois over $F$. There is, I am told, a "restriction map" Gal$(L_2/F)\rightarrow$Gal$(L_1/F)$. What I assume this means is that we restrict the action of $\sigma\in$ Gal$(L_2/F)$ to the field $L_1$, which would then be an automorphism of $L_1$. However, I am also aware that Gal$(L_1/F)$ is a quotient of Gal$(L_2/F)$, but I'm not sure if this is relevent. Is my interpretation correct?

Best Answer

  • If $\sigma \in Aut(L_1/F)$ and $L_2/ L_1$ is algebraic then (applying $\sigma$ to the coefficients of $L_1$-minimal polynomials) you can extend $\sigma$ to an isomorphism $L_2 \to \sigma(L_2)\subset\overline{L_2}$ fixing $F$

  • If $L_2/F$ is normal then $\sigma(L_2) = L_2$ so that $\sigma \in Aut(L_2/F)$

  • Conversely if $\sigma \in Aut(L_2/F)$ and $L_1/F$ is normal then $\sigma(L_1) = L_1$,

Thus the restriction map is surjective $Aut(L_2/F)\to Aut(L_1/F)$ which proves that $$Aut(L_1/F) = Aut(L_2/F)/\ker(\sigma \to \sigma|_{L_1}) = Aut(L_2/F)/Aut(L_2/L_1)$$

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