Need help in proving a corollary in Galois Theory related to Galois Group of Polynomial

Question

I am unable to prove a result in Galois Theory. The results is in Thomas Hungerford's Algebra. Can someone please tell me how to prove it? Unfortunately I have no clue how to prove it.

The result is ->

Please see Definition ->

Answer 1

By the Galois correspondence, the subgroup corresponding to $K(\Delta)$ consists of all automorphisms $\sigma$ of $L/K$ that fix $\Delta$. By proposition $4.5$ (can you see why this is true?), $\sigma(\Delta)=\Delta$ if and only if $\sigma$ is an even permutation (an element of $A_n$). Hence the subgroup is equal to $G\cap A_n$.

Can you now prove yourself that $G\subset A_n$ if and only if $\Delta\in K$? Use the Galois correspondence again..