How do I calculate the class number? What is the class number simply?

[Firepanda]

I'm doing the last part of iii)

My lecturer has given a definition of the class number, abstract stuff and hasn't actually explained it in simple terms to get my head around.

In particular in any example when it comes to calculating the class number h(K) she just goes 'Hence h(K) = 2' or similar.

I really have no idea how to land on what h(K) can possibly be, can anyone fill me in?

Thus, how do I show the class number is ≤3, and how would I go about that in part iv)?

 

[morphism]

Are you familiar with any sort of upperbound for the class number? (E.g. the Minkowski bound.)