January 17th
Today I learned a strengthening of Bauer's theorem, from Keith Conrad . For $K/\QQ$ a number field and extensions $L_1/K$ and $L_2/K$ with Galois closure $M/K,$ we have $\op{Spl}(L_1/K)=\op{Spl}(L_2/K)$ (up to a set of primes of density $0$) if and only if $L_2\subseteq L_1$ and\[\bigcup_\sigma\sigma\op{Gal}(L_1/K)\sigma^{-1}=\op{Gal}(M/L_2).\]\todo{finish this}