Abstract
Here we give a proof of the ambiguous class number formula for quadratic number fields and show that it implies the quadratic reciprocity law.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For integral ideals, the statement is trivial since then \(N\mathfrak a = (1)\) is equivalent to \(\mathfrak a = (1)\).
- 2.
Those who are familiar with the first principles of cohomology get the sequence for free: The trivial sequence
in which H denotes the group of nonzero fractional principal ideals, provides the long exact sequence
$$\displaystyle \begin{aligned}1 \longrightarrow E^G \longrightarrow (k^\times)^G \longrightarrow H^G \longrightarrow H^{1}(G,E) \longrightarrow H^1(G,k^\times),\end{aligned} $$from which the claim follows using Hilbert’s Theorem 90 (H 1(G, k ×) = 1), the periodicity H 1(G, A) ≃ H −1(G, A) for cyclic groups G, as well as \((k^\times )^G = {\mathbb Q}^\times \), E G = {±1} and \({\mathbb Q}^\times /E^G \simeq P\).
References
E. Benjamin, C. Snyder, Elements of order four in the narrow class group of real quadratic fields. J. Aust. Math. Soc. 100, 21–32 (2016)
F. Lemmermeyer, Relations in the 2-class group of quadratic number fields, J. Austr. Math. Soc. 93, 115–120 (2012)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Lemmermeyer, F. (2021). Ambiguous Ideal Classes and Quadratic Reciprocity. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-78652-6_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-78651-9
Online ISBN: 978-3-030-78652-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)