Lesson plan
class number problem
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n\geq 1 a complete list of imaginary quadratic fields \mathbb{Q}(\sqrt{d}) (for negative integers d ) having class number n . It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as d \to -\infty . The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
Article summary
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n\geq 1 a complete list of imaginary quadratic fields \mathbb{Q}(\sqrt{d}) (for negative integers d ) having class number n . It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as d \to -\infty . The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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