Arithmetic and lattices of imaginary quadratic rings

 

 

 

The maximal rings of quadratic integers of the imaginary quadratic fields are the subject of this paper.

 

More precisely, I wish to differentiate these rings of quadratic integers into two groups, namely:

 

 

 

The first group (1) of quadratic integers can be handled relatively briefly.

 

These quadratic integers are incongruent to 1 (mod 4) and have integral basis  with principal surd  for radicand .  They can be viewed geometrically as forming a rectangular lattice on the plane with fundamental region of height  and width 1.

 

The norm is the square of the distance between the element on the plane and the origin.  It can be derived from the integers of the quadratic fields not congruent to 1 (mod 4) by multiplying an arbitrary element by it’s conjugate, that is:

 

 

 

Further, the minimal polynomial can then easily be formed:

 

 

 

The norm, , of element , can also be derived geometrically from a simple application of the Pythagorean theorem as depicted in the diagram below.

 

 

Of course, when , this is the ring of Gaussian integers and the lattice has a fundamental region of a square.

 

Addition and subtraction of elements is performed in the expected manner:

 

 

 

[Visualization examples of complex addition… parallelogram vector addition.]

 

Multiplication is accomplished by a simple application of the Binomial theorem:

 

 

 

¹ I’ve chosen here not to simplify these formulas with the knowledge that .  So, remember that squaring .

 

Division of elements is found through “rationalizing the denominator” in the form of multiplication by an element’s conjugate over it’s conjugate.  The conjugate is formed by replacing  with it’s reflection across the real axis… that is, :

 

 

 

Hence, division is performed:

 

 

 

[Various visualizations of complex multiplication…]

 

 

The second group (2) of quadratic integers are somewhat more involved.

 

These quadratic integers are congruent to 1 (mod 4) and have integral basis  with principal surd  for radicand .  They can be viewed geometrically as forming a hexagonal, or triangular, lattice on the plane with fundamental triangular region of base  and width 1, as depicted in the diagram below:

 

 

Ignore the numbers on the axes and tick marks on this diagram as these quantifiers are arbitrary for the general ring of quadratic integers 1 (mod 4).

 

The norm is the square of the distance between the element on the plane and the origin.  It can be derived from the integers of the quadratic fields congruent to 1 (mod 4) by multiplying an arbitrary element by it’s conjugate, that is:

 

 

 

Further, the minimal polynomial can then easily be formed:

 

 

 

The norm, , of element , can also be derived geometrically from an application of the Law of Cosines, , as depicted in the diagram below for , the Eisenstein integers.  Note that all included angles for the Law of Cosines used below are either 60^° or 120^° and that  and .  Consequently, this simplifies the Law of Cosines to the norm  for the Eisenstein integers.

 

 

 

 

The Eisenstein integers are here depicted on the (fundamental) lattice of the Gaussian integers over the complex plane.  The red segments represent the shortest distance from the Eisenstein integer to the origin, or the hypotenuse of the triangle used for the Law of Cosines.  The blue segments represent the real and imaginary portions of the Eisenstein integers as laid out on the real and w-axes.  These are the other two sides to the triangle used for the Law of Cosines.

 

The Eisenstein integers, as well as the other quadratic integers, , are all partitioned into six sextants on the complex plane.  As you can see from the diagram below, the diagram above covers all the possible +,- combinations.  The +,- combination of coordinates takes the place of the sign of the angle in a scalar application of the Law of Cosines.  I believe you can convince yourself that the sign combinations will generate the same sign as would be generated by the cosine function for the appropriate reference angle in a scalar application.

 

 

 

There is some variation to consider for the general ring of quadratic integers, .

 

First of all, the lattice is stretched vertically, altering the expanses of the plane that each of these six sextant covers.  The angles for the Law of Cosines change, but there are only two angles for each ring of quadratic integers, .  The correction to the above norm equation for this is applied with the coefficient to the b term, .  This is unity with the Eisenstein integers.

 

[More algebraic and geometric detail on how this term operates on the norm equation]

 

The formulas for the basic arithmetic operations are little altered from the ones for quadratic integers, , except for a critical adjustment, and some changes resulting from this adjustment.

 

An adjustment must be made for the fact that the imaginary axis for the quadratic integers, , is not orthogonal, or perpendicular, to the real axis.  As one can see with application of the Law of Cosines, or perhaps more clearly through vector analysis of the dot product associated with it, multiplication by an element orthogonally reflected across the real axis, the conjugate, can only be expected to produce a scalar norm result if the two segments, or vectors, are perpendicular to one another.  However, because the basis for the imaginary quadratic integers, , is not an orthogonal one, one cannot expect a scalar norm result multiplying an element by its conjugate.

 

Consequently, the adjustment necessary is to replace all occurrences of  in arithmetic calculations with elements , with .

 

[More detailed explanation algebraically and geometrically]

 

Conjugation remains as it does for the Eisenstein integers, except it’s no longer true that for the general quadratic integers, , that the conjugate of an element can simply be found by replacing the principal surd with it’s reflection across the real-axis:

 

 

 

This doesn’t work because the principal surds associated with the other quadratic integers, , do not have norms equal to unity… as is the case with the Eisenstein integers.

 

[Explanation both algebraically and geometrically…]

 

However, the formula used to determine an element’s conjugate over the Eisenstein integers does work for the other quadratic integers, .  This is how one determines an element’s conjugate:

 

 

 

 

This diagram depicts the geometric construction of the conjugate of an Eisenstein integer laid over the (fundamental) lattice of the Gaussian integers on the complex plane.  The other imaginary quadratic integers, congruent 1 (mod 4) are similar except the imaginary axis’s angle of inclination to the real axis is greater.

 

Now,…

 

 

 

 

 

 

Change of basis formulas:

 

 

 

[cont…]