Editing Quadratic field (section)

== Class group ==
Determining the class group of a quadratic field extension can be accomplished using [[Minkowski's bound]] and the [[Kronecker symbol]] because of the finiteness of the [[Ideal class group|class group]].<ref>{{Cite web|title=Algebraic Number Theory, A Computational Approach|url=https://wstein.org/books/ant/ant.pdf|last=Stein|first=William|pages=77–86}}</ref> A quadratic field <math>K = \mathbf{Q}(\sqrt{d})</math> has [[Discriminant of an algebraic number field|discriminant]]
<math display=block>\Delta_K = \begin{cases}
d & d \equiv 1 \pmod 4 \\
4d & d \equiv 2,3 \pmod 4;
\end{cases}</math>
so the Minkowski bound is<ref>{{Cite web|last=Conrad|first=Keith|title=CLASS GROUP CALCULATIONS|url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgpex.pdf}}</ref><math display=block>M_K = \begin{cases}
2\sqrt{|\Delta|}/\pi & d < 0 \\
\sqrt{|\Delta|}/2 & d > 0 .
\end{cases}
 </math>

Then, the ideal class group is generated by the prime ideals whose norm is less than <math>M_K</math>. This can be done by looking at the decomposition of the ideals <math>(p)</math> for <math>p \in \mathbf{Z}</math> prime where <math>|p| < M_k.</math><ref name=":0" /> <sup>page 72</sup> These decompositions can be found using the [[Dedekind–Kummer theorem]].