Some questions about A classical introduction to modern number theory

Question

Recently, I'm studying the chapter12,A classical introduction to modern number theory, and I have some questions.

1. On page 172, there is this sentence' If L/K is separable, then t is not identically zero', why?
2. On the page 178, about the proof of theorem 1(The class number of F is finite), 'by proposition 12.2.3 M! can be contained in at most finitely many ideals', why? And why A~B can conclude $h_F<\infty$?

Answer 1

You really should provide more context rather than assume people know the notation without seeing the book.

Based just on what you wrote, I think your $t$ is the trace mapping $L \to K$. It is a very important property of finite separable extensions $L/K$ that the trace mapping is not identically $0$. In fact, $L/K$ is separable if and only if the trace mapping $L \to K$ is not identically $0$. See Theorem 1.1 here.

For your second question, if $M! \subset I$ for an ideal $I$, then $(M!) \subset I$ as ideals, so $I \mid (M!)$. Every nonzero ideal has only finitely ideal factors. See here for a discussion of how to deduce the finiteness of the class number.