Some Extension to Group Theory and Algebraic Number Theory

In this post, I will talk about the connection between continued fraction expansion of sqrt(d) (where d is a non-square positive integer) and the group of units in the ring of integers of a quadratic number field, namely Q(sqrt(d)), which consists of elements of the form a+b*sqrt(d), where a and b are rationals. 

There is a few special elements in the field Q(sqrt(d)),  the first one is the integral element, which is a root of a monic polynomial with integer coefficients. Note that a monic polynomial is a polynomial whose highest term has a coefficient of 1. For example, consider x = 2 - sqrt(5) in Q(sqrt(5)), and we can find the monic quadratic polynomial that x is a root of: 

x - 2 = sqrt(5)

(x − 2)2 = 5

x2 − 4x − 1 = 0

Since x2 − 4x − 1 is a monic polynomial, and x is one of its roots, we conclude that x is an integral element of Q(sqrt(5)).

Secondly, the ring of integers is the ring of all integral elements contained in Q(sqrt(d)), and we denote this ring by A. These rings have always been interests to mathematicians as studying them provides critical insights into the arithmetic properties of the number field, and opens up avenues for research in algebra, geometry and cryptography.

We also define a norm map from Q(sqrt(5)) to Z: we denote the norm of x = a + b*sqrt(d) by N(x), and we define N(x) = (a + b*sqrt(d)) (a - b*sqrt(d)) = a2 −db2.

Now let's switch to units in a ring. In simple terms, a unit is an element that has a multiplicative inverse which also lies in the ring. Here, the multiplication is not necessarily the usual multiplication between numbers - a ring has two operations, and since mathematicians are usually quite lazy, they have just named those two operations as addition and multiplication. But what is an inverse? In a ring, there is a multiplicative identity, which whenever multiplied by an element in the ring gives the element itself, so a unit is an element which when multiplied by another specific element in the ring, gives the multiplicative identity. But as an example, we will consider the ring of integers, and in this ring, the multiplication is just the elementary number multiplication! In the integers, it's not hard to see that the multiplicative identity is 1 and the only units are ±1. 

The norm map is a special map that maps the units in a number field to the units in the integers, which are ±1 in this case. 

One thing that we are particularly interested about the ring of integers, A, is the positive unit group contained in A. We call a unit positive when the unit's absolute value is greater or equal to 1. This group of positive units is cyclic, so we would like to find the generator of the group, which we call the fundamental unit. Does the name "fundamental" sound familiar? Yes! In here, the term "fundamental" also refers to the size of the element, in other words, all other positive units are powers of this fundamental unit. And guess what we use to find the fundamental unit? Continued fractions!

We can tell the general form of the elements in A take, depending on the modular arithmetic. I will show one example when d is congruent to 3 mod 4 (it's exactly the same for d congruent to 2 mod 4, and slightly more work for d congruent to 1 mod 4. Note we don't consider then case when d is a multiple of 4, because we still assume d is square free here!)

Consider Q(sqrt(7)), and we would like to find the group of positive units of the ring of integers. We have that sqrt(7) = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]. Recall from my last post I said that pk2 −dqk2 = ±1 when k is the second till last position in each repeat of the period. In the case of d = 7, we have the period length is 4.

Suppose the fundamental unit equals a + b*sqrt(7), and we would like to find the fundamental solution for the equation: a2 −7b2 = ±1. The first three convergents C4k−1 we get are 8/3, 127/48 and 2024/765. Therefore the first three units greater than 1 are 8+3*sqrt(7), 127+48*sqrt(7) and 2024+765*sqrt(7), and you can check that they are all powers of 8+3*sqrt(7). Thus the group of positive units is generated by 8+3*sqrt(7).