Abstract
This chapter introduces quadratic number fields, their rings of integers, and the group of rational and integral points on Pell conics and explains the connection with the technique of Vieta jumping.
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Notes
- 1.
In extensions with non-abelian Galois groups one has to distinguish carefully between these notations since στ(α) is often meant to be σ(τ(α)), whereas α στ = (α σ)τ.
- 2.
It is not clear a priori that such a maximal ring always exists.
- 3.
The discriminant of a quadratic number field does not depend on the choice of the integral basis; see Exercise 2.3.
- 4.
Binet published his formula in 1843; it was already known to Daniel Bernoulli in 1728—see [11, p. 90].
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Lemmermeyer, F. (2021). Quadratic Number Fields. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_2
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