Infrastructure: Structure Inside the Class Group of a Real Quadratic Field

The Unit Group of a Real Quadratic Field

While the unit group of an imaginary quadratic field is very simple, the unit group of a real quadratic field has nontrivial structure. Its study involves some geometry and analysis, but also it relates to Pell's equation and continued fractions, topics from elementary number theory.

Defining relations of a group $Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field

In this paper‎, ‎we have shown that the coset diagrams for the‎ ‎action of a linear-fractional group $Gamma$ generated by the linear-fractional‎ ‎transformations $r:zrightarrow frac{z-1}{z}$ and $s:zrightarrow frac{-1}{2(z+1)}$ on‎ ‎the rational projective line is connected and transitive‎. ‎By using coset diagrams‎, ‎we have shown that $r^{3}=s^{4}=1$ are defining relations for $Gamma$‎. ‎Furt...

The 4-class Group of Real Quadratic Number Fields

In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire’s result that the 2-class field tower of a real quadratic number field is infinite if its ideal class group has 4-rank ≥ 4, using a technique due to F. Hajir.

Computing the Hilbert class field of real quadratic fields

Using the units appearing in Stark’s conjectures on the values of L-functions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, so that k = Q( √ dk), and let ω denote an algebraic integer such that the ring of integers of k is Ok := Z+ ωZ. An important invariant of ...

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup