I just did the following exercise out of Neukirch's Algebraic Number Theory:
"A prime ideal p of K is totally split in the separable extension L|K iff it is totally split in the Galois closure N|K of L|K"
Now, I managed to solve this using a complicated argument using decomposition groups and it was not at all too pretty. Considering Neukirch introduces the decomposition group in the pages following this exercise, I suspect that there is a nicer way to prove this. So, my question is:
Are there any slick ways to show this? If so, how?