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

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SUMMARY

The class number, denoted as h(K), is a crucial concept in algebraic number theory that quantifies the failure of unique factorization in the ring of integers of a number field K. In the discussion, the participant seeks clarity on calculating h(K) and mentions the Minkowski bound as a potential upper limit for the class number. The lecturer's examples suggest that h(K) can be determined through specific calculations, but the participant requires a more straightforward explanation and methodology to derive these values. Understanding the Minkowski bound is essential for establishing upper limits on the class number.

PREREQUISITES
  • Basic knowledge of algebraic number theory
  • Understanding of number fields and their properties
  • Familiarity with the concept of unique factorization
  • Knowledge of the Minkowski bound in relation to class numbers
NEXT STEPS
  • Study the calculation methods for class numbers in algebraic number fields
  • Research the Minkowski bound and its application in determining class numbers
  • Explore examples of class number calculations for specific number fields
  • Learn about the relationship between class numbers and unique factorization domains
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Students of algebraic number theory, mathematicians interested in number fields, and anyone seeking to understand the implications of class numbers in unique factorization.

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I'm doing the last part of iii)

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

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