Computing the Unit Group of a Number Field

Starting from Pell’s equation and units in real quadratic fields, we will discuss the problem of computing the unit group of a number field. This will lead to a discussion of the Hidden Subgroup Problem, which arises in many quantum algorithms.  We will discuss recent work of Eisentrager, Hallgren, Kitaev, and Song, and of Biasse and Song, giving a quantum algorithm for this problem that runs in polynomial time.